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diff --git a/Docs/source/theory/AMR/AMR.tex b/Docs/source/theory/AMR/AMR.tex deleted file mode 100644 index 0c3ba520d..000000000 --- a/Docs/source/theory/AMR/AMR.tex +++ /dev/null @@ -1,76 +0,0 @@ -\input{newcommands} - -\begin{figure}[htb] - \centering - \includegraphics[width=15cm]{figures/ICNSP_2011_Vay_fig1.png} - \caption{Sketches of the implementation of mesh refinement in WarpX with the electrostatic (left) and electromagnetic (right) solvers. In both cases, the charge/current from particles are deposited at the finest levels first, then interpolated recursively to coarser levels. In the electrostatic case, the potential is calculated first at the coarsest level $L_0$, the solution interpolated to the boundaries of the refined patch $r$ at the next level $L_{1}$ and the potential calculated at $L_1$. The procedure is repeated iteratively up to the highest level. In the electromagnetic case, the fields are computed independently on each grid and patch without interpolation at boundaries. Patches are terminated by absorbing layers (PML) to prevent the reflection of electromagnetic waves. Additional coarse patch $c$ and fine grid $a$ are needed so that the full solution is obtained by substitution on $a$ as $F_{n+1}(a)=F_{n+1}(r)+I[F_n( s )-F_{n+1}( c )]$ where $F$ is the field, and $I$ is a coarse-to-fine interpolation operator. In both cases, the field solution at a given level $L_n$ is unaffected by the solution at higher levels $L_{n+1}$ and up, allowing for mitigation of some spurious effects (see text) by providing a transition zone via extension of the patches by a few cells beyond the desired refined area (red \& orange rectangles) in which the field is interpolated onto particles from the coarser parent level only.} - \label{fig:ESAMR} -\end{figure} - -The mesh refinement methods that have been implemented in WarpX were developed following the following principles: i) avoidance of spurious effects from mesh refinement, or minimization of such effects; ii) user controllability of the spurious effects' relative magnitude; iii) simplicity of implementation. The two main generic issues that were identified are: a) spurious self-force on macroparticles close to the mesh refinement interface \cite{Vaylpb2002,Colellajcp2010}; b) reflection (and possible amplification) of short wavelength electromagnetic waves at the mesh refinement interface \cite{Vayjcp01}. The two effects are due to the loss of translation invariance introduced by the asymmetry of the grid on each side of the mesh refinement interface. - -In addition, for some implementations where the field that is computed at a given level is affected by the solution at finer levels, there are cases where the procedure violates the integral of Gauss' Law around the refined patch, leading to long range errors \cite{Vaylpb2002,Colellajcp2010}. As will be shown below, in the procedure that has been developed in WarpX, the field at a given refinement level is not affected by the solution at finer levels, and is thus not affected by this type of error. - -\subsection{Electrostatic} -A cornerstone of the Particle-In-Cell method is that assuming a particle lying in a hypothetical infinite grid, then if the grid is regular and symmetrical, and if the order of field gathering matches the order of charge (or current) deposition, then there is no self-force of the particle acting on itself: a) anywhere if using the so-called ``momentum conserving'' gathering scheme; b) on average within one cell if using the ``energy conserving'' gathering scheme \cite{Birdsalllangdon}. A breaking of the regularity and/or symmetry in the grid, whether it is from the use of irregular meshes or mesh refinement, and whether one uses finite difference, finite volume or finite elements, results in a net spurious self-force (which does not average to zero over one cell) for a macroparticle close to the point of irregularity (mesh refinement interface for the current purpose) \cite{Vaylpb2002,Colellajcp2010}. - -A sketch of the implementation of mesh refinement in WarpX is given in Figure~\ref{fig:ESAMR} (left). Given the solution of the electric potential at a refinement level $L_n$, it is interpolated onto the boundaries of the grid patch(es) at the next refined level $L_{n+1}$. The electric potential is then computed at level $L_{n+1}$ by solving the Poisson equation. This procedure necessitates the knowledge of the charge density at every level of refinement. For efficiency, the macroparticle charge is deposited on the highest level patch that contains them, and the charge density of each patch is added recursively to lower levels, down to the lowest. - -\begin{figure}[htb] - \centering - \includegraphics[width=15cm]{figures/ICNSP_2011_Vay_fig2.png} - \caption{Position history of one charged particle attracted by its image induced by a nearby metallic (dirichlet) boundary. The particle is initialized at rest. Without refinement patch (reference case), the particle is accelerated by its image, is reflected specularly at the wall, then decelerates until it reaches its initial position at rest. If the particle is initialized inside a refinement patch, the particle is initially accelerated toward the wall but is spuriously reflected before it reaches the boundary of the patch whether using the method implemented in WarpX or the MC method. Providing a surrounding transition region 2 or 4 cells wide in which the potential is interpolated from the parent coarse solution reduces significantly the effect of the spurious self-force. } - \label{fig:ESselfforce} -\end{figure} -The presence of the self-force is illustrated on a simple test case that was introduced in \cite{Vaylpb2002} and also used in \cite{Colellajcp2010}: a single macroparticle is initialized at rest within a single refinement patch four cells away from the patch refinement boundary. The patch at level $L_1$ has $32\times32$ cells and is centered relative to the lowest $64\times64$ grid at level $L_0$ (``main grid''), while the macroparticle is centered in one direction but not in the other. The boundaries of the main grid are perfectly conducting, so that the macroparticle is attracted to the closest wall by its image. Specular reflection is applied when the particle reaches the boundary so that the motion is cyclic. The test was performed with WarpX using either linear or quadratic interpolation when gathering the main grid solution onto the refined patch boundary. It was also performed using another method from P. McCorquodale et al (labeled ``MC'' in this paper) based on the algorithm given in \cite{Mccorquodalejcp2004}, which employs a more elaborate procedure involving two-ways interpolations between the main grid and the refined patch. A reference case was also run using a single $128\times128$ grid with no refined patch, in which it is observed that the particle propagates toward the closest boundary at an accelerated pace, is reflected specularly at the boundary, then slows down until it reaches its initial position at zero velocity. The particle position histories are shown for the various cases in Fig. \ref{fig:ESselfforce}. In all the cases using the refinement patch, the particle was spuriously reflected near the patch boundary and was effectively trapped in the patch. We notice that linear interpolation performs better than quadratic, and that the simple method implemented in WarpX performs better than the other proposed method for this test (see discussion below). - -\begin{figure}[htb] - \centering - \includegraphics[width=15cm]{figures/ICNSP_2011_Vay_fig3.png} - \caption{(left) Maps of the magnitude of the spurious self-force $\epsilon$ in arbitrary units within one quarter of the refined patch, defined as $\epsilon=\sqrt{(E_x-E_x^{ref})^2+(E_y-E_y^{ref})^2}$, where $E_x$ and $E_y$ are the electric field components within the patch experienced by one particle at a given location and $E_x^{ref}$ and $E_y^{ref}$ are the electric field from a reference solution. The map is given for the WarpX and the MC mesh refinement algorithms and for linear and quadratic interpolation at the patch refinement boundary. \\(right) Lineouts of the maximum (taken over neighboring cells) of the spurious self-force. Close to the interface boundary (x=0), the spurious self-force decreases at a rate close to one order of magnitude per cell (red line), then at about one order of magnitude per six cells (green line).} - \label{fig:ESselfforcemap} -\end{figure} -The magnitude of the spurious self-force as a function of the macroparticle position was mapped and is shown in Fig. \ref{fig:ESselfforcemap} for the WarpX and MC algorithms using linear or quadratic interpolations between grid levels. It is observed that the magnitude of the spurious self-force decreases rapidly with the distance between the particle and the refined patch boundary, at a rate approaching one order of magnitude per cell for the four cells closest to the boundary and about one order of magnitude per six cells beyond. The method implemented in WarpX offers a weaker spurious force on average and especially at the cells that are the closest to the coarse-fine interface where it is the largest and thus matters most. -We notice that the magnitude of the spurious self-force depends strongly on the distance to the edge of the patch and to the nodes of the underlying coarse grid, but weakly on the order of deposition and size of the patch. - -A method was devised and implemented in WarpX for reducing the magnitude of spurious self-forces near the coarse-fine boundaries as follows. Noting that the coarse grid solution is unaffected by the presence of the patch and is thus free of self-force, extra ``transition'' cells are added around the ``effective'' refined area. -Within the effective area, the particles gather the potential in the fine grid. In the extra transition cells surrounding the refinement patch, the force is gathered directly from the coarse grid (an option, which has not yet been implemented, would be to interpolate between the coarse and fine grid field solutions within the transition zone so as to provide continuity of the force experienced by the particles at the interface). The number of cells allocated in the transition zones is controllable by the user in WarpX, giving the opportunity to check whether the spurious self-force is affecting the calculation by repeating it using different thicknesses of the transition zones. The control of the spurious force using the transition zone is illustrated in Fig.~\ref{fig:ESselfforce}, where the calculation with WarpX using linear interpolation at the patch interface was repeated using either two or four cells transition regions (measured in refined patch cell units). Using two extra cells allowed for the particle to be free of spurious trapping within the refined area and follow a trajectory that is close to the reference one, and using four extra cells improved further to the point where the resulting trajectory becomes undistinguishable from the reference one. -We note that an alternative method was devised for reducing the magnitude of self-force near the coarse-fine boundaries for the MC method, by using a special deposition procedure near the interface \cite{Colellajcp2010}. - -%\begin{figure}[htb] -% \centering -% \includegraphics[width=15cm]{figures/ICNSP_2011_Vay_fig4.png} -% \caption{Snapshot from a 3D self-consistent simulation of the injector in the High Current Experiment shows the beam emerging from the source at low energy (blue) and being accelerated (green-yellow-orange) and transported in a four quadrupole front end. The automatic layout of the mesh refinement patches from a 2D axisymmetric simulation of the source area shows 2 levels of refinement, concentrating the finer meshes around the emitter (white curve surface) and the beam edge (dark blue).} -% \label{fig:ESHCX} -%\end{figure} -%Automatic remeshing has been implemented in WarpX following the procedure described in \cite{Vaynim2005}, refining on criteria based on measures of local charge density magnitude and gradients. AMR WarpX simulations were applied to the modeling of the front end injector of the High Current Experiment (HCX) \cite{Prostprstab2005}, and provided the first numerically converged estimates of phase space beam distorsions, which directly affects beam quality \cite{Vaypop04}. Fig.~\ref{fig:ESHCX} shows snapshots from 2D axisymmetric simulation of the souce area illustrating the automatic placement of refined patches, and 3D simulation of the full injector showing the beam generation, acceleration and transport. - -\subsection{Electromagnetic} -The method that is used for electrostatic mesh refinement is not directly applicable to electromagnetic calculations. As was shown in section 3.4 of \cite{Vayjcp01}, refinement schemes relying solely on interpolation between coarse and fine patches lead to the reflection with amplification of the short wavelength modes that fall below the cutoff of the Nyquist frequency of the coarse grid. Unless these modes are damped heavily or prevented from occurring at their source, they may affect particle motion and their effect can escalate if trapped within a patch, via multiple successive reflections with amplification. - -To circumvent this issue, an additional coarse patch (with the same resolution as the parent grid) is added, as shown in Fig.~\ref{fig:ESAMR}-right and described in \cite{Vaycpc04}. Both the fine and the coarse grid patches are terminated by Perfectly Matched Layers, reducing wave reflection by orders of magnitude, controllable by the user \cite{Berengerjcp96,Vayjcp02}. The source current resulting from the motion of charged macroparticles within the refined region is accumulated on the fine patch and is then interpolated onto the coarse patch and added onto the parent grid. The process is repeated recursively from the finest level down to the coarsest. The Maxwell equations are then solved for one time interval on the entire set of grids, by default for one time step using the time step of the finest grid. The field on the coarse and fine patches only contain the contributions from the particles that have evolved within the refined area but not from the current sources outside the area. The total contribution of the field from sources within and outside the refined area is obtained by adding the field from the refined grid $F(r)$, and adding an interpolation $I$ of the difference between the relevant subset $s$ of the field in the parent grid $F(s)$ and the field of the coarse grid $F( c )$, on an auxiliary grid $a$, i.e. $F(a)=F(r)+I[F(s)-F( c )]$. The field on the parent grid subset $F(s)$ contains contributions from sources from both within and outside of the refined area. Thus, in effect, there is substitution of the coarse field resulting from sources within the patch area by its fine resolution counterpart. The operation is carried out recursively starting at the coarsest level up to the finest. -An option has been implemented in which various grid levels are pushed with different time steps, given as a fixed fraction of the individual grid Courant conditions (assuming same cell aspect ratio for all grids and refinement by integer factors). In this case, the fields from the coarse levels, which are advanced less often, are interpolated in time. - -The substitution method has two potential drawbacks due to the inexact cancellation between the coarse and fine patches of : (i) the remnants of ghost fixed charges created by the particles entering and leaving the patches (this effect is due to the use of the electromagnetic solver and is different from the spurious self-force that was described for the electrostatic case); (ii) if using a Maxwell solver with a low-order stencil, the electromagnetic waves traveling on each patch at slightly different velocity due to numerical dispersion. -The first issue results in an effective spurious multipole field whose magnitude decreases very rapidly with the distance to the patch boundary, similarly to the spurious self-force in the electrostatic case. Hence, adding a few extra transition cells surrounding the patches mitigates this effect very effectively. -%[Add hyperbolic correction?] -The tunability of WarpX's electromagnetic finite-difference and pseudo-spectral solvers provides the means to optimize the numerical dispersion so as to minimize the second effect for a given application, which has been demonstrated on the laser-plasma interaction test case presented in \cite{Vaycpc04}. -Both effects and their mitigation are described in more detail in \cite{Vaycpc04}. - -Caustics are supported anywhere on the grid with an accuracy that is set by the local resolution, and will be adequately resolved if the grid resolution supports the necessary modes from their sources to the points of wavefront crossing. The mesh refinement method that is implemented in WarpX has the potential to provide higher efficiency than the standard use of fixed gridding, by offering a path toward adaptive gridding following wavefronts. - -%\begin{figure}[htb] -% \centering -% \includegraphics[width=13cm]{figures/ICNSP_2011_Vay_fig5.png} -% \caption{Electron density $n_e$ (normalized to the density of the injected plasma) from WarpX simulations in 2-1/2D for a), b), c) and 3D for d) of a rigid beam (thin light-blue outline) propagating through a neutral plasma, for grid sizes of a) $128\times320$, b) $512\times1280$, c) $128\times320$ (main grid, red box) + $128\times640$ (patch 1, orange box) + $128\times1280$ (patch 2, yellow box), such that the resolution of patch 2 matched the resolution of the grid used for b), d) grid size of $64\times64\times160$ (main grid, red box) + $64\times64\times320$ (patch 1, orange box) + $64\times64\times640$ (patch 2, yellow box). For c) and d), the number and weight of injected plasma macroparticles was adjusted to keep the number of macroparticles per cell constant in each grid at injection in front of the beam.} -% \label{fig:EMplasma} -%\end{figure} -%As a test to the electromagnetic PIC implementation, WarpX simulations of wave excitations by a beam propagating through plasma, as described in \cite{Kaganovichpop2004}, were conducted. In these simulations, a hard-edged, elliptical, rigid beam propagates at constant velocity $v_z = 0.5c$ where $c$ is the speed of light through an initially cold neutral plasma of initial density $n_0$. The beam has a flat-top density profile of $n_b = n_0/2$, and an elliptical shape of length $l = 15 c/\omega_p$ and diameter $d = l/10$, where $\omega_p$ is the electron plasma frequency. It is shown in \cite{Kaganovichpop2004} that waves with a wavenumber of approximately $2\omega_p/v_z$ are generated in the plasma by the beam's electrostatic field, and have larger amplitude inside the beam, due to their interaction with the beam's sharp edges. - -%Resolving the beam edge and the small structures developing in the wake inside the beam forces small cell sizes. The resolution that is needed for macroscopic convergence was explored in 2-1/2D in a series of four runs where the number of grid cells was varied from $64\times160$ to $512\times1280$ by incremental factors of 2. Third order spline interpolation was used for the beam and plasma macroparticle current deposition and force gathering. The details of the plasma wake were very similar between the two highest resolution cases, indicating that macroscopic convergence was reached. The results from the runs using $128\times320$ and $512\times1280$ grids are shown in Fig.~\ref{fig:EMplasma}. The result from the highest resolution run serves as the reference for subsequent calculations with mesh refinement. - -%A run was conducted where the main grid had $128\times320$ cells and was complemented by two refinement patches (with successive refinement factors of 2 in each direction), such that the resolution in the central patch matched the resolution of the case of reference. -%The number and weight of the injected plasma macroparticles was varied, such that the number of macroparticles per cell in each grid at injection was constant. Results are plotted in Fig.~\ref{fig:EMplasma} (bottom-left) showing a good reproduction of the fine scale structures within the central fine patch in good agreement with the reference case. -%Lastly, a three-dimensional simulation with mesh refinement of the same physical setup was conducted. The grid setup and 3D isosurfaces of the plasma electron density as the beam enters the plasma are shown in Fig.~\ref{fig:EMplasma} (bottom-right). As expected, structures similar to the ones observed in 2D are present within the beam envelope. The speedup achieved by the use of mesh refinement was estimated to be approximately one order of magnitude in 3D. - - diff --git a/Docs/source/theory/Boosted_frame.png b/Docs/source/theory/Boosted_frame.png Binary files differnew file mode 100644 index 000000000..792562c32 --- /dev/null +++ b/Docs/source/theory/Boosted_frame.png diff --git a/Docs/source/theory/Deposition/Current_deposition.tex b/Docs/source/theory/Deposition/Current_deposition.tex deleted file mode 100644 index 18fedd807..000000000 --- a/Docs/source/theory/Deposition/Current_deposition.tex +++ /dev/null @@ -1,67 +0,0 @@ -\input{../newcommands} - -The current densities are deposited on the computational grid from -the particle position and velocities, employing splines of various -orders \cite{Abejcp86}. - -\begin{subequations} -\begin{eqnarray} -\rho & = & \frac{1}{\Delta x \Delta y \Delta z}\sum_nq_nS_n\\ -\mathbf{J} & = & \frac{1}{\Delta x \Delta y \Delta z}\sum_nq_n\mathbf{v_n}S_n -\end{eqnarray} -\end{subequations} - -In most applications, it is essential to prevent the accumulation -of errors resulting from the violation of the discretized Gauss' Law. -This is accomplished by providing a method for depositing the current -from the particles to the grid that preserves the discretized Gauss' -Law, or by providing a mechanism for ``divergence cleaning'' \cite{Birdsalllangdon,Langdoncpc92,Marderjcp87,Vaypop98,Munzjcp2000}. -For the former, schemes that allow a deposition of the current that -is exact when combined with the Yee solver is given in \cite{Villasenorcpc92} -for linear splines and in \cite{Esirkepovcpc01} for splines of arbitrary order. - -The NSFDTD formulations given above and in \cite{PukhovJPP99,Vayjcp2011,CowanPRSTAB13,LehePRSTAB13} -apply to the Maxwell-Faraday -equation, while the discretized Maxwell-Ampere equation uses the FDTD -formulation. Consequently, the charge conserving algorithms developed -for current deposition \cite{Villasenorcpc92,Esirkepovcpc01} apply -readily to those NSFDTD-based formulations. More details concerning -those implementations, including the expressions for the numerical -dispersion and Courant condition are given -in \cite{PukhovJPP99,Vayjcp2011,CowanPRSTAB13,LehePRSTAB13}. - -In the case of the pseudospectral solvers, the current deposition -algorithm generally does not satisfy the discretized continuity equation -in Fourier space $\tilde{\rho}^{n+1}=\tilde{\rho}^{n}-i\Delta t\fk\cdot\mathbf{\tilde{J}}^{n+1/2}$. -In this case, a Boris correction \cite{Birdsalllangdon} can be applied -in $k$ space in the form $\fe_{c}^{n+1}=\fe^{n+1}-\left(\fk\cdot\fe^{n+1}+i\tilde{\rho}^{n+1}\right)\fkhat/k$, -where $\fe_{c}$ is the corrected field. Alternatively, a correction -to the current can be applied (with some similarity to the current -deposition presented by Morse and Nielson in their potential-based -model in \cite{Morsenielson1971}) using $\fj_{c}^{n+1/2}=\fj^{n+1/2}-\left[\fk\cdot\fj^{n+1/2}-i\left(\tilde{\rho}^{n+1}-\tilde{\rho}^{n}\right)/\Delta t\right]\fkhat/k$, -where $\fj_{c}$ is the corrected current. In this case, the transverse -component of the current is left untouched while the longitudinal -component is effectively replaced by the one obtained from integration -of the continuity equation, ensuring that the corrected current satisfies -the continuity equation. The advantage of correcting the current rather than -the electric field is that it is more local and thus more compatible with -domain decomposition of the fields for parallel computation \cite{VayJCP2013}. - -Alternatively, an exact current deposition can be written for the pseudospectral solvers, following the geometrical interpretation of existing methods in real space \cite{Morsenielson1971,Villasenorcpc92,Esirkepovcpc01}, thereby averaging the currents of the paths following grid lines between positions $(x^n,y^n)$ and $(x^{n+1},y^{n+1})$, which is given in 2D (extension to 3D follows readily) for $k\neq0$ by \cite{VayJCP2013}: -% -\begin{eqnarray} -\fj^{k\neq0}=\frac{i\mathbf{\tilde{D}}}{\fk} -\label{Eq_Jdep_1} -\end{eqnarray} -with -\begin{eqnarray} -D_x = \frac{1}{2\Delta t}\sum_i q_i - [\Gamma(x_i^{n+1},y_i^{n+1})-\Gamma(x_i^{n},y_i^{n+1}) \nonumber\\ -+\Gamma(x_i^{n+1},y_i^{n})-\Gamma(x_i^{n},y_i^{n})],\\ -D_y = \frac{1}{2\Delta t}\sum_i q_i - [\Gamma(x_i^{n+1},y_i^{n+1})-\Gamma(x_i^{n+1},y_i^{n}) \nonumber \\ -+\Gamma(x_i^{n},y_i^{n+1})-\Gamma(x_i^{n},y_i^{n})], -\end{eqnarray} -where $\Gamma$ is the macro-particle form factor. -% -The contributions for $k=0$ are integrated directly in real space \cite{VayJCP2013}. diff --git a/Docs/source/theory/Gather/Field_gather.tex b/Docs/source/theory/Gather/Field_gather.tex deleted file mode 100644 index 18fedd807..000000000 --- a/Docs/source/theory/Gather/Field_gather.tex +++ /dev/null @@ -1,67 +0,0 @@ -\input{../newcommands} - -The current densities are deposited on the computational grid from -the particle position and velocities, employing splines of various -orders \cite{Abejcp86}. - -\begin{subequations} -\begin{eqnarray} -\rho & = & \frac{1}{\Delta x \Delta y \Delta z}\sum_nq_nS_n\\ -\mathbf{J} & = & \frac{1}{\Delta x \Delta y \Delta z}\sum_nq_n\mathbf{v_n}S_n -\end{eqnarray} -\end{subequations} - -In most applications, it is essential to prevent the accumulation -of errors resulting from the violation of the discretized Gauss' Law. -This is accomplished by providing a method for depositing the current -from the particles to the grid that preserves the discretized Gauss' -Law, or by providing a mechanism for ``divergence cleaning'' \cite{Birdsalllangdon,Langdoncpc92,Marderjcp87,Vaypop98,Munzjcp2000}. -For the former, schemes that allow a deposition of the current that -is exact when combined with the Yee solver is given in \cite{Villasenorcpc92} -for linear splines and in \cite{Esirkepovcpc01} for splines of arbitrary order. - -The NSFDTD formulations given above and in \cite{PukhovJPP99,Vayjcp2011,CowanPRSTAB13,LehePRSTAB13} -apply to the Maxwell-Faraday -equation, while the discretized Maxwell-Ampere equation uses the FDTD -formulation. Consequently, the charge conserving algorithms developed -for current deposition \cite{Villasenorcpc92,Esirkepovcpc01} apply -readily to those NSFDTD-based formulations. More details concerning -those implementations, including the expressions for the numerical -dispersion and Courant condition are given -in \cite{PukhovJPP99,Vayjcp2011,CowanPRSTAB13,LehePRSTAB13}. - -In the case of the pseudospectral solvers, the current deposition -algorithm generally does not satisfy the discretized continuity equation -in Fourier space $\tilde{\rho}^{n+1}=\tilde{\rho}^{n}-i\Delta t\fk\cdot\mathbf{\tilde{J}}^{n+1/2}$. -In this case, a Boris correction \cite{Birdsalllangdon} can be applied -in $k$ space in the form $\fe_{c}^{n+1}=\fe^{n+1}-\left(\fk\cdot\fe^{n+1}+i\tilde{\rho}^{n+1}\right)\fkhat/k$, -where $\fe_{c}$ is the corrected field. Alternatively, a correction -to the current can be applied (with some similarity to the current -deposition presented by Morse and Nielson in their potential-based -model in \cite{Morsenielson1971}) using $\fj_{c}^{n+1/2}=\fj^{n+1/2}-\left[\fk\cdot\fj^{n+1/2}-i\left(\tilde{\rho}^{n+1}-\tilde{\rho}^{n}\right)/\Delta t\right]\fkhat/k$, -where $\fj_{c}$ is the corrected current. In this case, the transverse -component of the current is left untouched while the longitudinal -component is effectively replaced by the one obtained from integration -of the continuity equation, ensuring that the corrected current satisfies -the continuity equation. The advantage of correcting the current rather than -the electric field is that it is more local and thus more compatible with -domain decomposition of the fields for parallel computation \cite{VayJCP2013}. - -Alternatively, an exact current deposition can be written for the pseudospectral solvers, following the geometrical interpretation of existing methods in real space \cite{Morsenielson1971,Villasenorcpc92,Esirkepovcpc01}, thereby averaging the currents of the paths following grid lines between positions $(x^n,y^n)$ and $(x^{n+1},y^{n+1})$, which is given in 2D (extension to 3D follows readily) for $k\neq0$ by \cite{VayJCP2013}: -% -\begin{eqnarray} -\fj^{k\neq0}=\frac{i\mathbf{\tilde{D}}}{\fk} -\label{Eq_Jdep_1} -\end{eqnarray} -with -\begin{eqnarray} -D_x = \frac{1}{2\Delta t}\sum_i q_i - [\Gamma(x_i^{n+1},y_i^{n+1})-\Gamma(x_i^{n},y_i^{n+1}) \nonumber\\ -+\Gamma(x_i^{n+1},y_i^{n})-\Gamma(x_i^{n},y_i^{n})],\\ -D_y = \frac{1}{2\Delta t}\sum_i q_i - [\Gamma(x_i^{n+1},y_i^{n+1})-\Gamma(x_i^{n+1},y_i^{n}) \nonumber \\ -+\Gamma(x_i^{n},y_i^{n+1})-\Gamma(x_i^{n},y_i^{n})], -\end{eqnarray} -where $\Gamma$ is the macro-particle form factor. -% -The contributions for $k=0$ are integrated directly in real space \cite{VayJCP2013}. diff --git a/Docs/source/theory/ICNSP_2011_Vay_fig1.png b/Docs/source/theory/ICNSP_2011_Vay_fig1.png Binary files differnew file mode 100644 index 000000000..ec687da7b --- /dev/null +++ b/Docs/source/theory/ICNSP_2011_Vay_fig1.png diff --git a/Docs/source/theory/ICNSP_2011_Vay_fig2.png b/Docs/source/theory/ICNSP_2011_Vay_fig2.png Binary files differnew file mode 100644 index 000000000..d03321f8e --- /dev/null +++ b/Docs/source/theory/ICNSP_2011_Vay_fig2.png diff --git a/Docs/source/theory/ICNSP_2011_Vay_fig3.png b/Docs/source/theory/ICNSP_2011_Vay_fig3.png Binary files differnew file mode 100644 index 000000000..d530b247c --- /dev/null +++ b/Docs/source/theory/ICNSP_2011_Vay_fig3.png diff --git a/Docs/source/theory/Input_output.png b/Docs/source/theory/Input_output.png Binary files differnew file mode 100644 index 000000000..daac48e61 --- /dev/null +++ b/Docs/source/theory/Input_output.png diff --git a/Docs/source/theory/Makefile b/Docs/source/theory/Makefile deleted file mode 100644 index f65bf6d58..000000000 --- a/Docs/source/theory/Makefile +++ /dev/null @@ -1,12 +0,0 @@ -SRC_FILES = theory.tex \ - Deposition/Current_deposition.tex \ - Gather/Field_gather.tex \ - Maxwell_solvers/Maxwell_FDTD_solver.tex \ - Maxwell_solvers/Maxwell_NSFDTD_solver.tex \ - Maxwell_solvers/Maxwell_PSATD_solver.tex - -all: $(SRC_FILES) - rm -f allbibs.bib;cat *.bib > allbibs.bib - rm -f theory.rst - pandoc theory.tex --mathjax -S --wrap=preserve --bibliography allbibs.bib -o theorycore.rst - cat theory_header.rst theorycore.rst > theory.rst diff --git a/Docs/source/theory/Maxwell_solvers/Maxwell_FDTD_solver.tex b/Docs/source/theory/Maxwell_solvers/Maxwell_FDTD_solver.tex deleted file mode 100644 index 1df65fe75..000000000 --- a/Docs/source/theory/Maxwell_solvers/Maxwell_FDTD_solver.tex +++ /dev/null @@ -1,38 +0,0 @@ -\input{../newcommands} - -The most popular algorithm for electromagnetic PIC codes is the Finite-Difference -Time-Domain (or FDTD) solver - -\begin{subequations} -\begin{eqnarray} -D_{t}\mathbf{B} & = & -\nabla\times\mathbf{E}\label{Eq:Faraday-2}\\ -D_{t}\mathbf{E} & = & \nabla\times\mathbf{B}-\mathbf{J}\label{Eq:Ampere-2}\\ -\left[\nabla\cdot\mathbf{E}\right. & = & \left.\rho\right]\label{Eq:Gauss-2}\\ -\left[\nabla\cdot\mathbf{B}\right. & = & \left.0\right].\label{Eq:divb-2} -\end{eqnarray} -\end{subequations} - -\begin{figure} -%\begin{centering} -\includegraphics[scale=0.7]{figures/Yee_grid.png} -%\par\end{centering} -\caption{\label{fig:yee_grid}(left) Layout of field components on the staggered ``Yee'' grid. Current densities and electric fields are defined on the edges of the cells and magnetic fields on the faces. (right) Time integration using a second-order finite-difference "leapfrog" integrator.} -\end{figure} - -The differential operator is defined as $\nabla=D_{x}\mathbf{\hat{x}}+D_{y}\mathbf{\hat{y}}+D_{z}\mathbf{\hat{z}}$ -and the finite-difference operators in time and space are defined -respectively as $ $$D_{t}G|_{i,j,k}^{n}=\left(G|_{i,j,k}^{n+1/2}-G|_{i,j,k}^{n-1/2}\right)/\Delta t$$ $ -and $D_{x}G|_{i,j,k}^{n}=\left(G|_{i+1/2,j,k}^{n}-G|_{i-1/2,j,k}^{n}\right)/\Delta x$, -where $\Delta t$ and $\Delta x$ are respectively the time step and -the grid cell size along $x$, $n$ is the time index and $i$, $j$ -and $k$ are the spatial indices along $x$, $y$ and $z$ respectively. -The difference operators along $y$ and $z$ are obtained by circular -permutation. The equations in brackets are given for completeness, -as they are often not actually solved, thanks to the usage of a so-called -charge conserving algorithm, as explained below. As shown in Figure -\ref{fig:yee_grid}, the quantities are given on a staggered (or ``Yee'') -grid \cite{Yee}, where the electric field components are located -between nodes and the magnetic field components are located in the -center of the cell faces. Knowing the current densities at half-integer steps, -the electric field components are updated alternately with the magnetic -field components at integer and half-integer steps respectively. diff --git a/Docs/source/theory/Maxwell_solvers/Maxwell_NSFDTD_solver.tex b/Docs/source/theory/Maxwell_solvers/Maxwell_NSFDTD_solver.tex deleted file mode 100644 index 770428be1..000000000 --- a/Docs/source/theory/Maxwell_solvers/Maxwell_NSFDTD_solver.tex +++ /dev/null @@ -1,57 +0,0 @@ -\input{../newcommands} - -In \cite{Coleieee1997,Coleieee2002}, Cole introduced an implementation -of the source-free Maxwell's wave equations for narrow-band applications -based on non-standard finite-differences (NSFD). In \cite{Karkicap06}, -Karkkainen \emph{et al.} adapted it for wideband applications. At -the Courant limit for the time step and for a given set of parameters, -the stencil proposed in \cite{Karkicap06} has no numerical dispersion -along the principal axes, provided that the cell size is the same -along each dimension (i.e. cubic cells in 3D). The ``Cole-Karkkainnen'' -(or CK) solver uses the non-standard finite difference formulation -(based on extended stencils) of the Maxwell-Ampere equation and can be -implemented as follows \cite{Vayjcp2011}: - -\begin{subequations} -\begin{eqnarray} -D_{t}\mathbf{B} & = & -\nabla^{*}\times\mathbf{E}\label{Eq:Faraday}\\ -D_{t}\mathbf{E} & = & \nabla\times\mathbf{B}-\mathbf{J}\label{Eq:Ampere}\\ -\left[\nabla\cdot\mathbf{E}\right. & = & \left.\rho\right]\label{Eq:Gauss}\\ -\left[\nabla^{*}\cdot\mathbf{B}\right. & = & \left.0\right]\label{Eq:divb} -\end{eqnarray} -\end{subequations} - -Eq. \ref{Eq:Gauss} and \ref{Eq:divb} are not being solved explicitly -but verified via appropriate initial conditions and current deposition -procedure. The NSFD differential operators is given by $\nabla^{*}=D_{x}^{*}\mathbf{\hat{x}}+D_{y}^{*}\mathbf{\hat{y}}+D_{z}^{*}\mathbf{\hat{z}}$ -where $D_{x}^{*}=\left(\alpha+\beta S_{x}^{1}+\xi S_{x}^{2}\right)D_{x}$ -with $S_{x}^{1}G|_{i,j,k}^{n}=G|_{i,j+1,k}^{n}+G|_{i,j-1,k}^{n}+G|_{i,j,k+1}^{n}+G|_{i,j,k-1}^{n}$, -$S_{x}^{2}G|_{i,j,k}^{n}=G|_{i,j+1,k+1}^{n}+G|_{i,j-1,k+1}^{n}+G|_{i,j+1,k-1}^{n}+G|_{i,j-1,k-1}^{n}$. -$G$ is a sample vector component, while $\alpha$, $\beta$ and $\xi$ -are constant scalars satisfying $\alpha+4\beta+4\xi=1$. As with -the FDTD algorithm, the quantities with half-integer are located between -the nodes (electric field components) or in the center of the cell -faces (magnetic field components). The operators along $y$ and $z$, -i.e. $D_{y}$, $D_{z}$, $D_{y}^{*}$, $D_{z}^{*}$, $S_{y}^{1}$, -$S_{z}^{1}$, $S_{y}^{2}$, and $S_{z}^{2}$, are obtained by circular -permutation of the indices. - -Assuming cubic cells ($\Delta x=\Delta y=\Delta z$), the coefficients -given in \cite{Karkicap06} ($\alpha=7/12$, $\beta=1/12$ and $\xi=1/48$) -allow for the Courant condition to be at $\Delta t=\Delta x$, which -equates to having no numerical dispersion along the principal axes. -The algorithm reduces to the FDTD algorithm with $\alpha=1$ and $\beta=\xi=0$. -An extension to non-cubic cells is provided by Cowan, \emph{et al.} -in 3-D in \cite{CowanPRSTAB13} and was given by Pukhov in 2-D in -\cite{PukhovJPP99}. An alternative NSFDTD implementation that enables superluminous waves is also -given by Lehe {\it et al.} in \cite{LehePRSTAB13}. - -As mentioned above, a key feature of the algorithms based on NSFDTD -is that some implementations \cite{Karkicap06,CowanPRSTAB13} enable the time step $\Delta t=\Delta x$ along one or -more axes and no numerical dispersion along those axes. However, as -shown in \cite{Vayjcp2011}, an instability develops at the Nyquist -wavelength at (or very near) such a timestep. It is also shown in -the same paper that removing the Nyquist component in all the source -terms using a bilinear filter (see description of the filter below) -suppresses this instability. - diff --git a/Docs/source/theory/Maxwell_solvers/Maxwell_PSATD_solver.tex b/Docs/source/theory/Maxwell_solvers/Maxwell_PSATD_solver.tex deleted file mode 100644 index 682651001..000000000 --- a/Docs/source/theory/Maxwell_solvers/Maxwell_PSATD_solver.tex +++ /dev/null @@ -1,87 +0,0 @@ -\input{../newcommands} - -Maxwell's equations in Fourier space are given by % --- Maxwell -\begin{subequations} -\begin{eqnarray} -\frac{\partial\fe}{\partial t} & = & i\fk\times\fb-\fj\\ -\frac{\partial\fb}{\partial t} & = & -i\fk\times\fe\\ -{}[i\fk\cdot\fe & = & \tilde{\rho}]\\ -{}[i\fk\cdot\fb & = & 0] -\end{eqnarray} -\end{subequations} -where $\tilde{a}$ is the Fourier Transform of the quantity $a$. -As with the real space formulation, provided that the continuity equation -$\partial\tilde{\rho}/\partial t+i\fk\cdot\fj=0$ is satisfied, then -the last two equations will automatically be satisfied at any time -if satisfied initially and do not need to be explicitly integrated. - -Decomposing the electric field and current between longitudinal and -transverse components $\fe=\fe_{L}+\fe_{T}=\fkhat(\fkhat\cdot\fe)-\fkhat\times(\fkhat\times\fe)$ -and $\fj=\fj_{L}+\fj_{T}=\fkhat(\fkhat\cdot\fj)-\fkhat\times(\fkhat\times\fj)$ -gives -\begin{subequations} -\begin{eqnarray} -\frac{\partial\fe_{T}}{\partial t} & = & i\fk\times\fb-\mathbf{\tilde{J}_{T}}\\ -\frac{\partial\fe_{L}}{\partial t} & = & -\mathbf{\tilde{J}_{L}}\\ -\frac{\partial\fb}{\partial t} & = & -i\fk\times\fe -\end{eqnarray} -\end{subequations} -with $\fkhat=\fk/k$. - -If the sources are assumed to be constant over a time interval $\Delta t$, -the system of equations is solvable analytically and is given by (see -\cite{Habericnsp73} for the original formulation and \cite{VayJCP13} -for a more detailed derivation): - -% --- PSATD -\begin{subequations} -\label{Eq:PSATD} -\begin{eqnarray} -\fe_{T}^{n+1} & = & C\fe_{T}^{n}+iS\fkhat\times\fb^{n}-\frac{S}{k}\fj_{T}^{n+1/2}\label{Eq:PSATD_transverse_1}\\ -\fe_{L}^{n+1} & = & \fe_{L}^{n}-\Delta t\fj_{L}^{n+1/2}\\ -\fb^{n+1} & = & C\fb^{n}-iS\fkhat\times\fe^{n}\\ -&+&i\frac{1-C}{k}\fkhat\times\fj^{n+1/2}\label{Eq:PSATD_transverse_2} -\end{eqnarray} -\end{subequations} -with $C=\cos\left(k\Delta t\right)$ and $S=\sin\left(k\Delta t\right)$. - -Combining the transverse and longitudinal components, gives -\begin{subequations} -\begin{eqnarray} -\fe^{n+1} & = & C\fe^{n}+iS\fkhat\times\fb^{n}-\frac{S}{k}\fj^{n+1/2}\\ - & + &(1-C)\fkhat(\fkhat\cdot\fe^{n})\nonumber \\ - & + & \fkhat(\fkhat\cdot\fj^{n+1/2})\left(\frac{S}{k}-\Delta t\right),\label{Eq_PSATD_1}\\ -\fb^{n+1} & = & C\fb^{n}-iS\fkhat\times\fe^{n}\\ -&+&i\frac{1-C}{k}\fkhat\times\fj^{n+1/2}.\label{Eq_PSATD_2} -\end{eqnarray} -\end{subequations} - -For fields generated by the source terms without the self-consistent -dynamics of the charged particles, this algorithm is free of numerical -dispersion and is not subject to a Courant condition. Furthermore, -this solution is exact for any time step size subject to the assumption -that the current source is constant over that time step. - -As shown in \cite{VayJCP13}, by expanding the coefficients $S_{h}$ -and $C_{h}$ in Taylor series and keeping the leading terms, the PSATD -formulation reduces to the perhaps better known pseudo-spectral time-domain -(PSTD) formulation \cite{DawsonRMP83,Liumotl1997}: % --- PSTD -\begin{subequations} -\begin{eqnarray} -\fe^{n+1} & = & \fe^{n}+i\Delta t\fk\times\fb^{n+1/2}-\Delta t\fj^{n+1/2},\\ -\fb^{n+3/2} & = & \fb^{n+1/2}-i\Delta t\fk\times\fe^{n+1}. -\end{eqnarray} -\end{subequations} -The dispersion relation of the PSTD solver is given by $\sin(\frac{\omega\Delta t}{2})=\frac{k\Delta t}{2}.$ -In contrast to the PSATD solver, the PSTD solver is subject to numerical -dispersion for a finite time step and to a Courant condition that -is given by $\Delta t\leq \frac{2}{\pi}\left(\frac{1}{\Delta x^{2}}+\frac{1}{\Delta y^{2}}+\frac{1}{\Delta x^{2}}\right)^{-1/2}.$ - -The PSATD and PSTD formulations that were just given apply to the -field components located at the nodes of the grid. As noted in \cite{Ohmurapiers2010}, -they can also be easily recast on a staggered Yee grid by multiplication -of the field components by the appropriate phase factors to shift -them from the collocated to the staggered locations. The choice between -a collocated and a staggered formulation is application-dependent. - -Spectral solvers used to be very popular in the years 1970s to early 1990s, before being replaced by finite-difference methods with the advent of parallel supercomputers that favored local methods. However, it was shown recently that standard domain decomposition with Fast Fourier Transforms that are local to each subdomain could be used effectively with PIC spectral methods \cite{VayJCP13}, at the cost of truncation errors in the guard cells that could be neglected. A detailed analysis of the effectiveness of the method with exact evaluation of the magnitude of the effect of the truncation error is given in \cite{Vincenti2016a} for stencils of arbitrary order (up-to the infinite ``spectral'' order). diff --git a/Docs/source/theory/PML/PML.log b/Docs/source/theory/PML/PML.log deleted file mode 100644 index e69de29bb..000000000 --- a/Docs/source/theory/PML/PML.log +++ /dev/null diff --git a/Docs/source/theory/PML/PML.tex b/Docs/source/theory/PML/PML.tex deleted file mode 100644 index 6956b5182..000000000 --- a/Docs/source/theory/PML/PML.tex +++ /dev/null @@ -1,222 +0,0 @@ -\input{newcommands} - -\subsection{Open boundary condition for electromagnetic waves} - -For the TE case, the original Berenger's Perfectly Matched Layer (PML) writes - -% PML -\begin{eqnarray} -\varepsilon _{0}\frac{\partial E_{x}}{\partial t}+\sigma _{y}E_{x} = & \frac{\partial H_{z}}{\partial y}\label{PML_def_1} \\ -\varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{\partial H_{z}}{\partial x}\label{PML_def_2} \\ -\mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{\partial E_{y}}{\partial x}\label{PML_def_3} \\ -\mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{\partial E_{x}}{\partial y}\label{PML_def_4} \\ -H_{z} = & H_{zx}+H_{zy}\label{PML_def_5} -\end{eqnarray} - -This can be generalized to - -% APML -\begin{eqnarray} -\varepsilon _{0}\frac{\partial E_{x}}{\partial t}+\sigma _{y}E_{x} = & \frac{c_{y}}{c}\frac{\partial H_{z}}{\partial y}+\overline{\sigma }_{y}H_{z}\label{APML_def_1} \\ -\varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{c_{x}}{c}\frac{\partial H_{z}}{\partial x}+\overline{\sigma }_{x}H_{z}\label{APML_def_2} \\ -\mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{c^{*}_{x}}{c}\frac{\partial E_{y}}{\partial x}+\overline{\sigma }_{x}^{*}E_{y}\label{APML_def_3} \\ -\mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{c^{*}_{y}}{c}\frac{\partial E_{x}}{\partial y}+\overline{\sigma }_{y}^{*}E_{x}\label{APML_def_4} \\ -H_{z} = & H_{zx}+H_{zy}\label{APML_def_5} -\end{eqnarray} - -For $c_{x}=c_{y}=c^{*}_{x}=c^{*}_{y}=c$ and $\overline{\sigma }_{x}=\overline{\sigma }_{y}=\overline{\sigma }_{x}^{*}=\overline{\sigma }_{y}^{*}=0$, -this system reduces to the Berenger PML medium, while adding the additional -constraint $\sigma _{x}=\sigma _{y}=\sigma _{x}^{*}=\sigma _{y}^{*}=0$ -leads to the system of Maxwell equations in vacuum. - -\subsubsection{\label{Sec:analytic theory, propa plane wave}Propagation of a Plane Wave in an APML Medium} - -We consider a plane wave of magnitude ($ E_{0},H_{zx0},H_{zy0} $) -and pulsation $\omega$ propagating in the APML medium with an -angle $\varphi$ relative to the x axis - -\begin{eqnarray} -E_{x} = & -E_{0}\sin \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_1} \\ -E_{y} = & E_{0}\cos \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_2} \\ -H_{zx} = & H_{zx0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_AMPL_def_3} \\ -H_{zy} = & H_{zy0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_4} -\end{eqnarray} - - -where $\alpha$ and$\beta$ are two complex constants to -be determined. - -Introducing (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_def_2}), -(\ref{Plane_wave_AMPL_def_3}) and (\ref{Plane_wave_APML_def_4}) -into (\ref{APML_def_1}), (\ref{APML_def_2}), (\ref{APML_def_3}) -and (\ref{APML_def_4}) gives - -\begin{eqnarray} -\varepsilon _{0}E_{0}\sin \varphi -i\frac{\sigma _{y}}{\omega }E_{0}\sin \varphi = & \beta \frac{c_{y}}{c}\left( H_{zx0}+H_{zy0}\right) +i\frac{\overline{\sigma }_{y}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_1} \\ -\varepsilon _{0}E_{0}\cos \varphi -i\frac{\sigma _{x}}{\omega }E_{0}\cos \varphi = & \alpha \frac{c_{x}}{c}\left( H_{zx0}+H_{zy0}\right) -i\frac{\overline{\sigma }_{x}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_2} \\ -\mu _{0}H_{zx0}-i\frac{\sigma ^{*}_{x}}{\omega }H_{zx0} = & \alpha \frac{c^{*}_{x}}{c}E_{0}\cos \varphi -i\frac{\overline{\sigma }^{*}_{x}}{\omega }E_{0}\cos \varphi \label{Plane_wave_APML_1_3} \\ -\mu _{0}H_{zy0}-i\frac{\sigma ^{*}_{y}}{\omega }H_{zy0} = & \beta \frac{c^{*}_{y}}{c}E_{0}\sin \varphi +i\frac{\overline{\sigma }^{*}_{y}}{\omega }E_{0}\sin \varphi \label{Plane_wave_APML_1_4} -\end{eqnarray} - - -Defining $Z=E_{0}/\left( H_{zx0}+H_{zy0}\right)$ and using (\ref{Plane_wave_APML_1_1}) -and (\ref{Plane_wave_APML_1_2}), we get - -\begin{eqnarray} -\beta = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi -i\frac{\overline{\sigma }_{y}}{\omega }\right] \frac{c}{c_{y}}\label{Plane_wave_APML_beta_of_g} \\ -\alpha = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi +i\frac{\overline{\sigma }_{x}}{\omega }\right] \frac{c}{c_{x}}\label{Plane_wave_APML_alpha_of_g} -\end{eqnarray} - - -Adding $H_{zx0}$ and $H_{zy0}$ from (\ref{Plane_wave_APML_1_3}) -and (\ref{Plane_wave_APML_1_4}) and substituting the expressions -for $\alpha$ and $\beta$ from (\ref{Plane_wave_APML_beta_of_g}) -and (\ref{Plane_wave_APML_alpha_of_g}) yields - -\begin{eqnarray} -\frac{1}{Z} = & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi \frac{c^{*}_{x}}{c_{x}}+i\frac{\overline{\sigma }_{x}}{\omega }\frac{c^{*}_{x}}{c_{x}}-i\frac{\overline{\sigma }^{*}_{x}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{x}}{\omega }}\cos \varphi \nonumber \\ - + & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi \frac{c^{*}_{y}}{c_{y}}-i\frac{\overline{\sigma }_{y}}{\omega }\frac{c^{*}_{y}}{c_{y}}+i\frac{\overline{\sigma }^{*}_{y}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{y}}{\omega }}\sin \varphi -\end{eqnarray} - - -If $c_{x}=c^{*}_{x}$, $c_{y}=c^{*}_{y}$, $\overline{\sigma }_{x}=\overline{\sigma }^{*}_{x}$, $\overline{\sigma }_{y}=\overline{\sigma }^{*}_{y}$, $\frac{\sigma _{x}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{x}}{\mu _{0}}$ and $\frac{\sigma _{y}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{y}}{\mu _{0}}$ then - -\begin{eqnarray} -Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance} -\end{eqnarray} - - -which is the impedance of vacuum. Hence, like the PML, given some -restrictions on the parameters, the APML does not generate any reflection -at any angle and any frequency. As for the PML, this property is not -retained after discretization, as shown subsequently in this paper. - -Calling $\psi$ any component of the field and $\psi _{0}$ -its magnitude, we get from (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_beta_of_g}), -(\ref{Plane_wave_APML_alpha_of_g}) and (\ref{APML_impedance}) that - -\begin{equation} -\label{Plane_wave_absorption} -\psi =\psi _{0}e^{i\omega \left( t\mp x\cos \varphi /c_{x}\mp y\sin \varphi /c_{y}\right) }e^{-\left( \pm \frac{\sigma _{x}\cos \varphi }{\varepsilon _{0}c_{x}}+\overline{\sigma }_{x}\frac{c}{c_{x}}\right) x}e^{-\left( \pm \frac{\sigma _{y}\sin \varphi }{\varepsilon _{0}c_{y}}+\overline{\sigma }_{y}\frac{c}{c_{y}}\right) y} -\end{equation} - - -We assume that we have an APML layer of thickness $\delta$ (measured -along $x$) and that $\sigma _{y}=\overline{\sigma }_{y}=0$ -and $c_{y}=c.$ Using (\ref{Plane_wave_absorption}), we determine -that the coefficient of reflection given by this layer is - -\begin{eqnarray} -R_{APML}\left( \theta \right) = & e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}+\overline{\sigma }_{x}c/c_{x}\right) \delta }e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}-\overline{\sigma }_{x}c/c_{x}\right) \delta }\nonumber \\ - = & e^{-2\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}\right) \delta } -\end{eqnarray} - - -which happens to be the same as the PML theoretical coefficient of -reflection if we assume $c_{x}=c$. Hence, it follows that for -the purpose of wave absorption, the term $\overline{\sigma }_{x}$ -seems to be of no interest. However, although this conclusion is true -at the infinitesimal limit, it does not hold for the discretized counterpart. - -\subsubsection{Discretization} - -% -\begin{subequations} -\begin{align} -\frac{E_x|^{n+1}_{j+1/2,k,l}-E_x|^{n}_{j+1/2,k,l}}{\Delta t} + \sigma_y \frac{E_x|^{n+1}_{j+1/2,k,l}+E_x|^{n}_{j+1/2,k,l}}{2} = & \frac{H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}}{\Delta y} \\ -% -\frac{E_y|^{n+1}_{j,k+1/2,l}-E_y|^{n}_{j,k+1/2,l}}{\Delta t} + \sigma_x \frac{E_y|^{n+1}_{j,k+1/2,l}+E_y|^{n}_{j,k+1/2,l}}{2} = & - \frac{H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}}{\Delta x} \\ -% -\frac{H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l}-H_{zx}|^{n}_{j+1/2,k+1/2,l}}{\Delta t} + \sigma^*_x \frac{H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l}+H_{zx}|^{n}_{j+1/2,k+1/2,l}}{2} = & - \frac{E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}}{\Delta x} \\ -% -\frac{H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l}-H_{zy}|^{n}_{j+1/2,k+1/2,l}}{\Delta t} + \sigma^*_y \frac{H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l}+H_{zy}|^{n}_{j+1/2,k+1/2,l}}{2} = & \frac{E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}}{\Delta y} \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} - -% -\begin{subequations} -\begin{align} -E_x|^{n+1}_{j+1/2,k,l} = & \left(\frac{1-\sigma_y \Delta t/2}{1+\sigma_y \Delta t/2}\right) E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t/\Delta y}{1+\sigma_y \Delta t/2} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ -% -E_y|^{n+1}_{j,k+1/2,l} = & \left(\frac{1-\sigma_x \Delta t/2}{1+\sigma_x \Delta t/2}\right) E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t/\Delta x}{1+\sigma_x \Delta t/2} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ -% -H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & \left(\frac{1-\sigma^*_x \Delta t/2}{1+\sigma^*_x \Delta t/2}\right) H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t/\Delta x}{1+\sigma^*_x \Delta t/2} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ -% -H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & \left(\frac{1-\sigma^*_y \Delta t/2}{1+\sigma^*_y \Delta t/2}\right) H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t/\Delta y}{1+\sigma^*_y \Delta t/2} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} - -% -\begin{subequations} -\begin{align} -E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} E_x|^{n}_{j+1/2,k,l} + \frac{1-e^{-\sigma_y\Delta t}}{\sigma_y \Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ -% -E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} E_y|^{n}_{j,k+1/2,l} - \frac{1-e^{-\sigma_x\Delta t}}{\sigma_x \Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ -% -H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{1-e^{-\sigma^*_x\Delta t}}{\sigma^*_x \Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ -% -H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{1-e^{-\sigma^*_y\Delta t}}{\sigma^*_y \Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} - - -% -\begin{subequations} -\begin{align} -E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} E_x|^{n}_{j+1/2,k,l} + \frac{1-e^{-\sigma_y\Delta t}}{\sigma_y \Delta y}\frac{c_y}{c} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ -% -E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} E_y|^{n}_{j,k+1/2,l} - \frac{1-e^{-\sigma_x\Delta t}}{\sigma_x \Delta x}\frac{c_x}{c} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ -% -H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{1-e^{-\sigma^*_x\Delta t}}{\sigma^*_x \Delta x}\frac{c^*_x}{c} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ -% -H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{1-e^{-\sigma^*_y\Delta t}}{\sigma^*_y \Delta y}\frac{c^*_y}{c} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} - - % -\begin{subequations} -\begin{align} -c_x = & c e^{-\sigma_x\Delta t} \frac{\sigma_x \Delta x}{1-e^{-\sigma_x\Delta t}} \\ -c_y = & c e^{-\sigma_y\Delta t} \frac{\sigma_y \Delta y}{1-e^{-\sigma_y\Delta t}} \\ -c^*_x = & c e^{-\sigma^*_x\Delta t} \frac{\sigma^*_x \Delta x}{1-e^{-\sigma^*_x\Delta t}} \\ -c^*_y = & c e^{-\sigma^*_y\Delta t} \frac{\sigma^*_y \Delta y}{1-e^{-\sigma^*_y\Delta t}} -\end{align} -\end{subequations} - -% -\begin{subequations} -\begin{align} -E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} \left[ E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t}{\Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \right] \\ -% -E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} \left[ E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \right] \\ -% -H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} \left[ H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \right] \\ -% -H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} \left[ H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t}{\Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \right] \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} - -% -\begin{subequations} -\begin{align} -E_x|^{n+1}_{j+1/2,k,l} = & E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t}{\Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ -% -E_y|^{n+1}_{j,k+1/2,l} = & E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ -% -H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ -% -H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t}{\Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ -% -H_z = & H_{zx}+H_{zy} -\end{align} -\end{subequations} diff --git a/Docs/source/theory/Particle_pushers/Boris_pusher.tex b/Docs/source/theory/Particle_pushers/Boris_pusher.tex deleted file mode 100644 index 56686d319..000000000 --- a/Docs/source/theory/Particle_pushers/Boris_pusher.tex +++ /dev/null @@ -1,26 +0,0 @@ -\input{../newcommands} - -The solution proposed by Boris \cite{BorisICNSP70} is given by -\begin{align} -\mathbf{\bar{v}}^{i}= & \frac{\gamma^{i+1/2}\mathbf{v}^{i+1/2}+\gamma^{i-1/2}\mathbf{v}^{i-1/2}}{2\bar{\gamma}^{i}}.\label{Eq:boris_v} -\end{align} -where $\bar{\gamma}^{i}$ is defined by $\bar{\gamma}^{i} \equiv (\gamma^{i+1/2}+\gamma^{i-1/2} )/2$. - -The system (\ref{Eq:leapfrog_v},\ref{Eq:boris_v}) is solved very -efficiently following Boris' method, where the electric field push -is decoupled from the magnetic push. Setting $\mathbf{u}=\gamma\mathbf{v}$, the -velocity is updated using the following sequence: - -\begin{subequations} -\begin{align} -\mathbf{u^{-}}= & \mathbf{u}^{i-1/2}+\left(q\Delta t/2m\right)\mathbf{E}^{i}\\ -\mathbf{u'}= & \mathbf{u}^{-}+\mathbf{u}^{-}\times\mathbf{t}\\ -\mathbf{u}^{+}= & \mathbf{u}^{-}+\mathbf{u'}\times2\mathbf{t}/(1+t^{2})\\ -\mathbf{u}^{i+1/2}= & \mathbf{u}^{+}+\left(q\Delta t/2m\right)\mathbf{E}^{i} -\end{align} -\end{subequations} -where $\mathbf{t}=\left(q\Delta - t/2m\right)\mathbf{B}^{i}/\bar{\gamma}^{i}$ and where -$\bar{\gamma}^{i}$ can be calculated as $\bar{\gamma}^{i}=\sqrt{1+(\mathbf{u}^-/c)^2}$. - -The Boris implementation is second-order accurate, time-reversible and fast. Its implementation is very widespread and used in the vast majority of PIC codes. diff --git a/Docs/source/theory/Particle_pushers/Vay_pusher.tex b/Docs/source/theory/Particle_pushers/Vay_pusher.tex deleted file mode 100644 index b2ed186d0..000000000 --- a/Docs/source/theory/Particle_pushers/Vay_pusher.tex +++ /dev/null @@ -1,26 +0,0 @@ -\input{newcommands} - -It was shown in \cite{Vaypop2008} that the Boris formulation is -not Lorentz invariant and can lead to significant errors in the treatment -of relativistic dynamics. A Lorentz invariant formulation is obtained -by considering the following velocity average -\begin{align} -\mathbf{\bar{v}}^{i}= & \frac{\mathbf{v}^{i+1/2}+\mathbf{v}^{i-1/2}}{2},\label{Eq:new_v} -\end{align} -This gives a system that is solvable analytically (see \cite{Vaypop2008} -for a detailed derivation), giving the following velocity update: - -\begin{subequations} -\begin{align} -\mathbf{u^{*}}= & \mathbf{u}^{i-1/2}+\frac{q\Delta t}{m}\left(\mathbf{E}^{i}+\frac{\mathbf{v}^{i-1/2}}{2}\times\mathbf{B}^{i}\right),\label{pusher_gamma}\\ -\mathbf{u}^{i+1/2}= & \left[\mathbf{u^{*}}+\left(\mathbf{u^{*}}\cdot\mathbf{t}\right)\mathbf{t}+\mathbf{u^{*}}\times\mathbf{t}\right]/\left(1+t^{2}\right),\label{pusher_upr} -\end{align} -\end{subequations} -where $\mathbf{t}=\bm{\tau}/\gamma^{i+1/2}$, $\bm{\tau}=\left(q\Delta t/2m\right)\mathbf{B}^{i}$, -$\gamma^{i+1/2}=\sqrt{\sigma+\sqrt{\sigma^{2}+\left(\tau^{2}+w^{2}\right)}}$, -$w=\mathbf{u^{*}}\cdot\bm{\tau}$, $\sigma=\left(\gamma'^{2}-\tau^{2}\right)/2$ -and $\gamma'=\sqrt{1+(\mathbf{u}^{*}/c)^{2}}$. This Lorentz invariant formulation -is particularly well suited for the modeling of ultra-relativistic -charged particle beams, where the accurate account of the cancellation -of the self-generated electric and magnetic fields is essential, as -shown in \cite{Vaypop2008}. diff --git a/Docs/source/theory/Plasma_acceleration_sim.png b/Docs/source/theory/Plasma_acceleration_sim.png Binary files differnew file mode 100644 index 000000000..d5c5fbf67 --- /dev/null +++ b/Docs/source/theory/Plasma_acceleration_sim.png diff --git a/Docs/source/theory/WarpX.bib b/Docs/source/theory/WarpX.bib deleted file mode 100644 index 38bf4c202..000000000 --- a/Docs/source/theory/WarpX.bib +++ /dev/null @@ -1,2227 +0,0 @@ -Automatically generated by Mendeley Desktop 1.16.1 -Any changes to this file will be lost if it is regenerated by Mendeley. - -BibTeX export options can be customized via Preferences -> BibTeX in Mendeley Desktop - -@article{QuickpicParallel, -author = {Feng, B. and Huang, C. and Decyk, V. and Mori, W.B. and Muggli, P. and Katsouleas, T.}, -doi = {10.1016/j.jcp.2009.04.019}, -issn = {00219991}, -journal = {Journal of Computational Physics}, -month = {aug}, -number = {15}, -pages = {5340--5348}, -title = {{Enhancing parallel quasi-static particle-in-cell simulations with a pipelining algorithm}}, -url = {http://linkinghub.elsevier.com/retrieve/pii/S002199910900206X}, -volume = {228}, -year = {2009} -} -@book{HockneyEastwoodBook, -author = {Hockney, R W and Eastwood, J W}, -isbn = {0-85274-392-0}, -pages = {xxi+540 pp}, -title = {{Computer simulation using particles}}, -type = {Book}, -year = {1988} -} -@article{Parkerjcp1991, -author = {Parker, Se and Birdsall, Ck}, -doi = {10.1016/0021-9991(91)90040-R}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {nov}, -number = {1}, -pages = {91--102}, -title = {{Numerical Error In Electron Orbits With Large Omega-Ce Delta-T}}, -volume = {97}, -year = {1991} -} -@inproceedings{Fawleyipac10, -address = {Kyoto, Japan}, -author = {Fawley, W M and Vay, J.-L.}, -booktitle = {Proc. Ipac 2010, Paper Tupec064}, -title = {{Full Electromagnetic Simulation Of Coherent Synchrotron Radiation Via The Lorentz-Boosted Frame Approach}}, -year = {2010} -} -@article{Godfreyjcp74, -author = {Godfrey, Bb}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -number = {4}, -pages = {504--521}, -title = {{Numerical Cherenkov Instabilities In Electromagnetic Particle Codes}}, -volume = {15}, -year = {1974} -} -@article{Quickpic, -author = {Huang, C and Decyk, V K and Ren, C and Zhou, M and Lu, W and Mori, W B and Cooley, J H and {Antonsen Jr.}, T M and Katsouleas, T}, -doi = {10.1016/J.Jcp.2006.01.039}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {sep}, -number = {2}, -pages = {658--679}, -title = {{Quickpic: A Highly Efficient Particle-In-Cell Code For Modeling Wakefield Acceleration In Plasmas}}, -volume = {217}, -year = {2006} -} -@article{Lundprstab2009, -author = {Lund, Steven M and Kikuchi, Takashi and Davidson, Ronald C}, -doi = {10.1103/Physrevstab.12.114801}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {nov}, -number = {11}, -title = {{Generation Of Initial Kinetic Distributions For Simulation Of Long-Pulse Charged Particle Beams With High Space-Charge Intensity}}, -volume = {12}, -year = {2009} -} -@article{CowanPRSTAB13, -author = {Cowan, Benjamin M and Bruhwiler, David L and Cary, John R and Cormier-Michel, Estelle and Geddes, Cameron G R}, -doi = {10.1103/PhysRevSTAB.16.041303}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {apr}, -number = {4}, -title = {{Generalized algorithm for control of numerical dispersion in explicit time-domain electromagnetic simulations}}, -volume = {16}, -year = {2013} -} -@article{Esareyrmp09, -author = {Esarey, E and Schroeder, C B and Leemans, W P}, -doi = {10.1103/Revmodphys.81.1229}, -issn = {0034-6861}, -journal = {Rev. Mod. Phys.}, -number = {3}, -pages = {1229--1285}, -title = {{Physics Of Laser-Driven Plasma-Based Electron Accelerators}}, -volume = {81}, -year = {2009} -} -@article{YuPRL2014, -author = {Yu, L.-L. and Esarey, E and Schroeder, C B and Vay, J.-L. and Benedetti, C and Geddes, C G R and Chen, M and Leemans, W P}, -doi = {10.1103/PhysRevLett.112.125001}, -journal = {Phys. Rev. Lett.}, -month = {mar}, -number = {12}, -pages = {125001}, -publisher = {American Physical Society}, -title = {{Two-Color Laser-Ionization Injection}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.112.125001}, -volume = {112}, -year = {2014} -} -@article{Vaynim2007, -annote = {16Th International Symposium On Heavy Ion Inertial Fusion, St Malo, France, Jul 09-14, 2006}, -author = {Vay, J.-L. and Furman, M A and Seidl, P A and Cohen, R H and Friedman, A and Grote, D P and Covo, M Kireeff and Molvik, A W and Stoltz, P H and Veitzer, S and Verboncoeur, J P}, -doi = {10.1016/J.Nima.2007.02.013}, -institution = {Univ Paris Sud Xi}, -issn = {0168-9002}, -journal = {Nuclear Instruments {\&} Methods In Physics Research Section A-Accelerators Spectrometers Detectors And Associated Equipment}, -month = {jul}, -number = {1-2}, -pages = {65--69}, -title = {{Self-Consistent Simulations Of Heavy-Ion Beams Interacting With Electron-Clouds}}, -volume = {577}, -year = {2007} -} -@article{Vay2000, -abstract = {A new absorbing boundary condition using an absorbing layer is presented for application to finite-difference time-domain (FDTD) calculation of the wave equation. This algorithm is by construction a hybrid between the Berenger perfectly matched layer (PML) algorithm and the one-way Sommerfeld algorithm. The new prescription contains both of these earlier ones as particular cases, and retains benefits from both. Numerical results indicate that the new algorithm provides absorbing rates superior to those of the PML algorithm.}, -author = {Vay, Jean-Luc}, -journal = {Journal of Computational Physics}, -keywords = {ABC,Berenger,PML,Sommerfeld,absorbing boundary condition,electromagnetic,finite-difference,perfectly matched layer,wave equation}, -number = {2}, -pages = {511--521}, -title = {{A New Absorbing Layer Boundary Condition for the Wave Equation}}, -url = {http://www.sciencedirect.com/science/article/pii/S0021999100966233}, -volume = {165}, -year = {2000} -} -@article{Vayjcp2011, -author = {Vay, J L and Geddes, C G R and Cormier-Michel, E and Grote, D P}, -doi = {10.1016/J.Jcp.2011.04.003}, -journal = {Journal of Computational Physics}, -month = {jul}, -number = {15}, -pages = {5908--5929}, -title = {{Numerical Methods For Instability Mitigation In The Modeling Of Laser Wakefield Accelerators In A Lorentz-Boosted Frame}}, -volume = {230}, -year = {2011} -} -@article{Krallpre1993, -author = {Krall, J and Ting, A and Esarey, E and Sprangle, P}, -doi = {10.1103/Physreve.48.2157}, -journal = {Physical Review E}, -month = {sep}, -number = {3}, -pages = {2157--2161}, -title = {{Enhanced Acceleration In A Self-Modulated-Laser Wake-Field Accelerator}}, -volume = {48}, -year = {1993} -} -@article{Vayarxiv10_2, -author = {{Vay $\backslash$It Et Al.}, J.-L.}, -journal = {Arxiv:1011.0917V2}, -title = {{Effects Of Hyperbolic Rotation In Minkowski Space On The Modeling Of Plasma Accelerators In A Lorentz Boosted Frame}}, -year = {2010} -} -@article{Dennisw1997585, -author = {W., Dennis and Hewett}, -doi = {10.1006/Jcph.1997.5835}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -number = {2}, -pages = {585--616}, -title = {{The Embedded Curved Boundary Method For Orthogonal Simulation Meshes}}, -url = {Http://www.sciencedirect.com/Science/Article/Pii/S0021999197958356}, -volume = {138}, -year = {1997} -} -@article{Habib2016, -abstract = {This draft report summarizes and details the findings, results, and recommendations derived from the ASCR/HEP Exascale Requirements Review meeting held in June, 2015. The main conclusions are as follows. 1) Larger, more capable computing and data facilities are needed to support HEP science goals in all three frontiers: Energy, Intensity, and Cosmic. The expected scale of the demand at the 2025 timescale is at least two orders of magnitude -- and in some cases greater -- than that available currently. 2) The growth rate of data produced by simulations is overwhelming the current ability, of both facilities and researchers, to store and analyze it. Additional resources and new techniques for data analysis are urgently needed. 3) Data rates and volumes from HEP experimental facilities are also straining the ability to store and analyze large and complex data volumes. Appropriately configured leadership-class facilities can play a transformational role in enabling scientific discovery from these datasets. 4) A close integration of HPC simulation and data analysis will aid greatly in interpreting results from HEP experiments. Such an integration will minimize data movement and facilitate interdependent workflows. 5) Long-range planning between HEP and ASCR will be required to meet HEP's research needs. To best use ASCR HPC resources the experimental HEP program needs a) an established long-term plan for access to ASCR computational and data resources, b) an ability to map workflows onto HPC resources, c) the ability for ASCR facilities to accommodate workflows run by collaborations that can have thousands of individual members, d) to transition codes to the next-generation HPC platforms that will be available at ASCR facilities, e) to build up and train a workforce capable of developing and using simulations and analysis to support HEP scientific research on next-generation systems.}, -archivePrefix = {arXiv}, -arxivId = {1603.09303}, -author = {Habib, Salman and Roser, Robert and Gerber, Richard and Antypas, Katie and Riley, Katherine and Williams, Tim and Wells, Jack and Straatsma, Tjerk and Almgren, A. and Amundson, J. and Bailey, S. and Bard, D. and Bloom, K. and Bockelman, B. and Borgland, A. and Borrill, J. and Boughezal, R. and Brower, R. and Cowan, B. and Finkel, H. and Frontiere, N. and Fuess, S. and Ge, L. and Gnedin, N. and Gottlieb, S. and Gutsche, O. and Han, T. and Heitmann, K. and Hoeche, S. and Ko, K. and Kononenko, O. and LeCompte, T. and Li, Z. and Lukic, Z. and Mori, W. and Nugent, P. and Ng, C. -K. and Oleynik, G. and O'Shea, B. and Padmanabhan, N. and Petravick, D. and Petriello, F. 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St-Ab}, -title = {{Propagation Of Higher Order Modes In Plasma Channels And Shaping Of The Transverse Field In Laser Plasma Accelerators}} -} -@article{Yee, -author = {Yee, Ks}, -issn = {0018-926X}, -journal = {Ieee Transactions On Antennas And Propagation}, -number = {3}, -pages = {302--307}, -title = {{Numerical Solution Of Initial Boundary Value Problems Involving Maxwells Equations In Isotropic Media}}, -volume = {Ap14}, -year = {1966} -} -@article{GodfreyJCP2013, -author = {Godfrey, Brendan B and Vay, Jean-Luc}, -doi = {http://dx.doi.org/10.1016/j.jcp.2013.04.006}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -keywords = {Numerical stability}, -number = {0}, -pages = {33--46}, -title = {{Numerical stability of relativistic beam multidimensional {\{}PIC{\}} simulations employing the Esirkepov algorithm}}, -url = {http://www.sciencedirect.com/science/article/pii/S0021999113002556}, -volume = {248}, -year = {2013} -} -@inproceedings{Cormieraac08, -author = {Cormier-Michel, E and Geddes, C G R and Esarey, E and Schroeder, C B and Bruhwiler, D L and Paul, K and Cowan, B and Leemans, W P}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -pages = {297--302}, -title = {{Scaled Simulations Of A 10 Gev Accelerator}}, -volume = {1086}, -year = {2009} -} -@inproceedings{Bruhwileraac08, -author = {Bruhwiler, D L and Cary, J R and Cowan, B M and Paul, K and Geddes, C G R and Mullowney, P J and Messmer, P and Esarey, E and Cormier-Michel, E and Leemans, W and Vay, J.-L.}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -pages = {29--37}, -title = {{New Developments In The Simulation Of Advanced Accelerator Concepts}}, -volume = {1086}, -year = {2009} -} -@article{Vaynim2005, -annote = {15Th International Symposium On Heavy Ion Inertial Fusion, Princeton, Nj, Jun 07-11, 2004}, -author = {Vay, Jl and Friedman, A and Grote, Dp}, -doi = {10.1016/J.Nima.2005.01.232}, -issn = {0168-9002}, -journal = {Nuclear Instruments {\&} Methods In Physics Research Section A-Accelerators Spectrometers Detectors And Associated Equipment}, -month = {may}, -number = {1-2}, -pages = {347--352}, -title = {{Application Of Adaptive Mesh Refinement To Pic Simulations In Heavy Ion Fusion}}, -volume = {544}, -year = {2005} -} -@article{BulanovSV2014, -abstract = {The paper examines the prospects of using laser plasma as a source of high-energy ions for the purpose of hadron beam therapy — an approach which is based on both theory and experimental results (ions are routinely observed to be accelerated in the interaction of high-power laser radiation with matter). Compared to therapy accelerators like synchrotrons and cyclotrons, laser technology is advantageous in that it is more compact and is simpler in delivering ions from the accelerator to the treatment room. Special target designs allow radiation therapy requirements for ion beam quality to be satisfied.}, -author = {{Bulanov S V and Wilkens J J and Esirkepov T Zh and Korn G and Kraft G and Kraft S D and Molls M and Khoroshkov V S}}, -issn = {1063-7869}, -journal = {Physics-Uspekhi}, -number = {12}, -pages = {1149}, -title = {{Laser ion acceleration for hadron therapy}}, -url = {http://stacks.iop.org/1063-7869/57/i=12/a=1149}, -volume = {57}, -year = {2014} -} -@article{Furmanprstab2002, -author = {Furman, Ma and Pivi, Mtf}, -doi = {10.1103/Physrevstab.5.124404}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {dec}, -number = {12}, -title = {{Probabilistic Model For The Simulation Of Secondary Electron Emission}}, -volume = {5}, -year = {2002} -} -@article{Qiang2014, -author = {Qiang, J. and Corlett, J. and Mitchell, C. E. and Papadopoulos, C. F. and Penn, G. and Placidi, M. and Reinsch, M. and Ryne, R. D. and Sannibale, F. and Sun, C. and Venturini, M. and Emma, P. and Reiche, S.}, -doi = {10.1103/PhysRevSTAB.17.030701}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Qiang et al. - 2014 - Start-to-end simulation of x-ray radiation of a next generation light source using the real number of electrons.pdf:pdf}, -issn = {1098-4402}, -journal = {Physical Review Special Topics - Accelerators and Beams}, -month = {mar}, -number = {3}, -pages = {030701}, -publisher = {American Physical Society}, -title = {{Start-to-end simulation of x-ray radiation of a next generation light source using the real number of electrons}}, -url = {http://link.aps.org/doi/10.1103/PhysRevSTAB.17.030701}, -volume = {17}, -year = {2014} -} -@inproceedings{Fawleyaac08, -author = {Fawley, W M and Vay, J.-L.}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -pages = {346--350}, -title = {{Use Of The Lorentz-Boosted Frame Transformation To Simulate Free-Electron Laser Amplifier Physics}}, -volume = {1086}, -year = {2009} -} -@misc{Bruhwilerpc08, -annote = {Private Communication}, -author = {Bruhwiler, D L}, -title = {{No Title}}, -year = {2008} -} -@article{Yu2014, -abstract = {Simulating laser wakefield acceleration (LWFA) in a Lorentz boosted frame in which the plasma drifts towards the laser with nu(b) can speed up the simulation by factors of gamma(2)(b) = (1 nu(2)(b)/c(2))(-1). In these simulations the relativistic drifting plasma inevitably induces a high frequency numerical instability that contaminates the interesting physics. Various approaches have been proposed to mitigate this instability. One approach is to solve Maxwell equations in Fourier space (a spectral solver) as this has been shown to suppress the fastest growing modes of this instability in simple test problems using a simple low pass or "ring" or "shell" like filters in Fourier space. We describe the development of a fully parallelized, multi-dimensional, particle-in-cell code that uses a spectral solver to solve Maxwell's equations and that includes the ability to launch a laser using a moving antenna. This new EM-PIC code is called UPIC-EMMA and it is based on the components of the UCLA PIC framework (UPIC). We show that by using UPIC-EMMA, LWFA simulations in the boosted frames with arbitrary yb can be conducted without the presence of the numerical instability. We also compare the results of a few LWFA cases for several values of yb, including lab frame simulations using OSIRIS, an EM-PIC code with a finite-difference time domain (FDTD) Maxwell solver. These comparisons include cases in both linear and nonlinear regimes. We also investigate some issues associated with numerical dispersion in lab and boosted frame simulations and between FDTD and spectral solvers. (C) 2014 Elsevier Inc. All rights reserved.}, -author = {Yu, Peicheng and Xu, Xinlu and Decyk, Viktor K. and An, Weiming and Vieira, Jorge and Tsung, Frank S. and Fonseca, Ricardo A. and Lu, Wei and Silva, Luis O. and Mori, Warren B.}, -doi = {10.1016/j.jcp.2014.02.016}, -issn = {00219991}, -journal = {Journal of Computational Physics}, -keywords = {Boosted frame simulation,IN-CELL CODE,INSTABILITIES,Laser wakefield accelerator,Numerical Cerenkov instability,PARTICLE SIMULATION,PLASMAS,Particle-in-cell,Plasma simulation,Spectral solver}, -month = {jun}, -pages = {124--138}, -publisher = {ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA}, -title = {{Modeling of laser wakefield acceleration in Lorentz boosted frame using EM-PIC code with spectral solver}}, -url = {https://apps.webofknowledge.com/full{\_}record.do?product=UA{\&}search{\_}mode=GeneralSearch{\&}qid=2{\&}SID=1CanLFIHrQ5v8O7cxqV{\&}page=1{\&}doc=5}, -volume = {266}, -year = {2014} -} -@article{Vaypop2011, -author = {Vay, J -L. and Geddes, C G R and Esarey, E and Schroeder, C B and Leemans, W P and Cormier-Michel, E and Grote, D P}, -doi = {10.1063/1.3663841}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {dec}, -number = {12}, -title = {{Modeling Of 10 Gev-1 Tev Laser-Plasma Accelerators Using Lorentz Boosted Simulations}}, -volume = {18}, -year = {2011} -} -@article{LehePRSTAB13, -author = {Lehe, R and Lifschitz, A and Thaury, C and Malka, V and Davoine, X}, -doi = {10.1103/PhysRevSTAB.16.021301}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {feb}, -number = {2}, -title = {{Numerical growth of emittance in simulations of laser-wakefield acceleration}}, -volume = {16}, -year = {2013} -} -@article{Langdoncpc92, -author = {Langdon, A B}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -month = {jul}, -number = {3}, -pages = {447--450}, -title = {{On Enforcing Gauss Law In Electromagnetic Particle-In-Cell Codes}}, -volume = {70}, -year = {1992} -} -@article{Colellajcp2010, -author = {Colella, Phillip and Norgaard, Peter C}, -doi = {10.1016/J.Jcp.2009.07.004}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {feb}, -number = {4}, -pages = {947--957}, -title = {{Controlling Self-Force Errors At Refinement Boundaries For Amr-Pic}}, -volume = {229}, -year = {2010} -} -@article{Coleieee2002, -author = {Cole, Jb}, -doi = {10.1109/Tap.2002.801268}, -issn = {0018-926X}, -journal = {Ieee Transactions On Antennas And Propagation}, -month = {sep}, -number = {9}, -pages = {1185--1191}, -title = {{High-Accuracy Yee Algorithm Based On Nonstandard Finite Differences: New Developments And Verifications}}, -volume = {50}, -year = {2002} -} -@article{HajimaNIM09, -annote = {Workshop on Compton Sources for X-gamma Rays, Porto Conte, ITALY, SEP 07-12, 2008}, -author = {Hajima, R and Kikuzawa, N and Nishimori, N and Hayakawa, T and Shizuma, T and Kawase, K and Kando, M and Minehara, E and Toyokawa, H and Ohgaki, H}, -doi = {10.1016/j.nima.2009.05.063}, -institution = {Ist Nazl Fis Nucl; ICFA}, -issn = {0168-9002}, -journal = {NUCLEAR INSTRUMENTS {\&} METHODS IN PHYSICS RESEARCH SECTION A-ACCELERATORS SPECTROMETERS DETECTORS AND ASSOCIATED EQUIPMENT}, -month = {sep}, -number = {1}, -pages = {S57--S61}, -title = {{Detection of radioactive isotopes by using laser Compton scattered gamma-ray beams}}, -volume = {608}, -year = {2009} -} -@article{MatlisJOSA11, -author = {Matlis, N H and Plateau, G R and van Tilborg, J and Leemans, W P}, -doi = {10.1364/JOSAB.28.000023}, -issn = {0740-3224}, -journal = {JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS}, -month = {jan}, -number = {1}, -pages = {23--27}, -title = {{Single-shot spatiotemporal measurements of ultrashort THz waveforms using temporal electric-field cross correlation}}, -volume = {28}, -year = {2011} -} -@article{Quickpic2, -abstract = {We present improvements to the three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm, which is used to efficiently model short-pulse laser and particle beam–plasma interactions. In this algorithm the fields including the index of refraction created by a static particle/laser beam are calculated. These fields are then used to advance the particle/laser beam forward in time (distance). For a 3D quasi-static code, calculating the wake fields is done using a two-dimensional (2D) PIC code where the time variable is $\xi$=ct-z and z is the propagation direction of the particle/laser beam. When calculating the wake, the fields, particle positions and momenta are not naturally time centered so an iterative predictor corrector loop is required. In the previous iterative loop in QuickPIC (currently the only 3D quasi-static PIC code), the field equations are derived using the Lorentz gauge. Here we describe a new algorithm which uses gauge independent field equations. It is found that with this new algorithm, the results converge to the results from fully explicitly PIC codes with far fewer iterations (typically 1 iteration as compared to 2–8) for a wide range of problems. In addition, we describe a new deposition scheme for directly depositing the time derivative of the current that is needed in one of the field equations. The new deposition scheme does not require message passing for the particles inside the iteration loop, which greatly improves the speed for parallelized calculations. Comparisons of results from the new and old algorithms and to fully explicit PIC codes are also presented.}, -author = {An, Weiming and Decyk, Viktor K. and Mori, Warren B. and Antonsen, Thomas M.}, -doi = {10.1016/j.jcp.2013.05.020}, -issn = {00219991}, -journal = {Journal of Computational Physics}, -pages = {165--177}, -title = {{An improved iteration loop for the three dimensional quasi-static particle-in-cell algorithm: QuickPIC}}, -volume = {250}, -year = {2013} -} -@misc{Cowanpriv2010, -annote = {Private Communication}, -author = {Cowan, B}, -title = {{No Title}}, -year = {2010} -} -@inproceedings{Geddesscidac09, -author = {{Geddes $\backslash$It Et Al.}, Cgr}, -booktitle = {Scidac Review 13}, -pages = {13}, -title = {{Laser Plasma Particle Accelerators: Large Fields For Smaller Facility Sources}}, -year = {2009} -} -@article{Gomberoffpop2007, -author = {Gomberoff, K and Fajans, J and Friedman, A and Grote, D and Vay, J.-L. and Wurtele, J S}, -doi = {10.1063/1.2778420}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {oct}, -number = {10}, -title = {{Simulations Of Plasma Confinement In An Antihydrogen Trap}}, -volume = {14}, -year = {2007} -} -@article{Morapop1997, -author = {Mora, P and Antonsen, Tm}, -doi = {10.1063/1.872134}, -issn = {1070-664X}, -journal = {Phys. Plasmas}, -month = {jan}, -number = {1}, -pages = {217--229}, -title = {{Kinetic Modeling Of Intense, Short Laser Pulses Propagating In Tenuous Plasmas}}, -volume = {4}, -year = {1997} -} -@inproceedings{Cowanaac08, -author = {Cowan, B and Bruhwiler, D and Cormier-Michel, E and Esarey, E and Geddes, C G R and Messmer, P and Paul, K}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -pages = {309--314}, -title = {{Laser Wakefield Simulation Using A Speed-Of-Light Frame Envelope Model}}, -volume = {1086}, -year = {2009} -} -@article{Cohenprstab2009, -annote = {17Th International Symposium On Heavy Ion Inertial Fusion, Tokyo, Japan, Aug 04-08, 2008}, -author = {Cohen, R H and Friedman, A and Grote, D P and Vay, J -L.}, -doi = {10.1016/J.Nima.2009.03.083}, -institution = {Tokyo Inst Technol, Res Lab Nucl Reactors; Japan Soc Plasma Sci {\&} Nucl Fus Res; Particle Accelerator Soc Japan}, -issn = {0168-9002}, -journal = {Nuclear Instruments {\&} Methods In Physics Research Section A-Accelerators Spectrometers Detectors And Associated Equipment}, -month = {jul}, -number = {1-2}, -pages = {53--55}, -title = {{An Implicit ``Drift-Lorentz{\{}''{\}} Mover For Plasma And Beam Simulations}}, -volume = {606}, -year = {2009} -} -@article{Kaganovich2012, -abstract = {Neutralized drift compression offers an effective means for particle beam pulse compression and current amplification. In neutralized drift compression, a linear longitudinal velocity tilt (head-to-tail gradient) is applied to the non-relativistic beam pulse, so that the beam pulse compresses as it drifts in the focusing section. The beam current can increase by more than a factor of 100 in the longitudinal direction. We have performed an analytical study of how errors in the velocity tilt acquired by the beam in the induction bunching module limit the maximum longitudinal compression. It is found that the compression ratio is determined by the relative errors in the velocity tilt. That is, one-percent errors may limit the compression to a factor of one hundred. However, a part of the beam pulse where the errors are small may compress to much higher values, which are determined by the initial thermal spread of the beam pulse. It is also shown that sharp jumps in the compressed current density profile can be produced due to overlaying of different parts of the pulse near the focal plane. Examples of slowly varying and rapidly varying errors compared to the beam pulse duration are studied. For beam velocity errors given by a cubic function, the compression ratio can be described analytically. In this limit, a significant portion of the beam pulse is located in the broad wings of the pulse and is poorly compressed. The central part of the compressed pulse is determined by the thermal spread. The scaling law for maximum compression ratio is derived. In addition to a smooth variation in the velocity tilt, fast-changing errors during the pulse may appear in the induction bunching module if the voltage pulse is formed by several pulsed elements. Different parts of the pulse compress nearly simultaneously at the target and the compressed profile may have many peaks. The maximum compression is a function of both thermal spread and the velocity errors. The effects of the finite gap width of the bunching module on compression are analyzed analytically. {\textcopyright} 2012 Elsevier B.V. All rights reserved.}, -author = {Kaganovich, Igor D. and Massidda, Scott and Startsev, Edward A. and Davidson, Ronald C. and Vay, Jean Luc and Friedman, Alex}, -journal = {Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment}, -keywords = {Beam dynamics,Longitudinal compression,Voltage errors}, -pages = {48--63}, -title = {{Effects of errors in velocity tilt on maximum longitudinal compression during neutralized drift compression of intense beam pulses: I. General description}}, -volume = {678}, -year = {2012} -} -@article{Vincenti2016, -author = {Vincenti, H. and Lehe, R. and Sasanka, R. and Vay, J-L.}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Vincenti et al. - 2016 - An efficient and portable SIMD algorithm for chargecurrent deposition in Particle-In-Cell codes.pdf:pdf}, -journal = {Computer Programs in Physics}, -pages = {To appear}, -title = {{An efficient and portable SIMD algorithm for charge/current deposition in Particle-In-Cell codes}}, -url = {https://arxiv.org/abs/1601.02056}, -year = {2016} -} -@article{Vaypop2008, -author = {Vay, J L}, -doi = {10.1063/1.2837054}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -pages = {56701}, -title = {{Simulation Of Beams Or Plasmas Crossing At Relativistic Velocity}}, -volume = {15}, -year = {2008} -} -@article{Wieland2016, -author = {Wieland, Volkmar and Pohl, Martin and Niemiec, Jacek and Rafighi, Iman and Nishikawa, Ken-Ichi}, -doi = {10.3847/0004-637X/820/1/62}, -issn = {1538-4357}, -journal = {The Astrophysical Journal}, -month = {mar}, -number = {1}, -pages = {62}, -title = {{NONRELATIVISTIC PERPENDICULAR SHOCKS MODELING YOUNG SUPERNOVA REMNANTS: NONSTATIONARY DYNAMICS AND PARTICLE ACCELERATION AT FORWARD AND REVERSE SHOCKS}}, -url = {http://stacks.iop.org/0004-637X/820/i=1/a=62?key=crossref.df89490cada654d013db835d06161d02}, -volume = {820}, -year = {2016} -} -@misc{Vay, -author = {Vay, J.-L.}, -title = {{Traditional HPC needs: particle accelerators}}, -url = {http://www.nersc.gov/assets/Uploads/DOEExascaleReviewVay.pdf} -} -@article{Faurenature04, -author = {Faure, J and Glinec, Y and Pukhov, A and Kiselev, S and Gordienko, S and Lefebvre, E and Rousseau, Jp and Burgy, F and Malka, V}, -doi = {10.1038/Nature02963}, -issn = {0028-0836}, -journal = {Nature}, -month = {sep}, -number = {7008}, -pages = {541--544}, -title = {{A Laser-Plasma Accelerator Producing Monoenergetic Electron Beams}}, -volume = {431}, -year = {2004} -} -@article{Greenwoodjcp04, -author = {Greenwood, Ad and Cartwright, Kl and Luginsland, Jw and Baca, Ea}, -doi = {10.1016/J.Jcp.2004.06.021}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {dec}, -number = {2}, -pages = {665--684}, -title = {{On The Elimination Of Numerical Cerenkov Radiation In Pic Simulations}}, -volume = {201}, -year = {2004} -} -@article{GodfreyIEEE2014, -abstract = {The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC) algorithm solves the vacuum Maxwell's equations exactly, has no Courant time-step limit (as conventionally defined), and offers substantial flexibility in plasma and particle beam simulations. It is, however, not free of the usual numerical instabilities, including the numerical Cherenkov instability, when applied to relativistic beam simulations. This paper presents several approaches that, when combined with digital filtering, almost completely eliminate the numerical Cherenkov instability. The paper also investigates the numerical stability of the PSATD algorithm at low beam energies.}, -author = {Godfrey, Brendan B. and Vay, Jean Luc and Haber, Irving}, -journal = {IEEE Transactions on Plasma Science}, -keywords = {Accelerators,numerical stability,particle beams,particle-in-cell (PIC),relativistic effects,simulation,spectral methods}, -number = {5}, -pages = {1339--1344}, -publisher = {Institute of Electrical and Electronics Engineers Inc.}, -title = {{Numerical stability improvements for the pseudospectral EM PIC algorithm}}, -volume = {42}, -year = {2014} -} -@inproceedings{Cowanicap09, -address = {San Francisco, Ca}, -author = {Cowan, B}, -booktitle = {Poster {\$}10{\^{}}{\{}Th{\}}{\$} Internat. Comput. Accel. Phys. Conf.}, -title = {{No Title}}, -year = {2009} -} -@article{Wuprstab2011, -author = {Wu, H -C. and Meyer-Ter-Vehn, J and Hegelich, B M and Fernandez, J C}, -doi = {10.1103/Physrevstab.14.070702}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {jul}, -number = {7}, -title = {{Nonlinear Coherent Thomson Scattering From Relativistic Electron Sheets As A Means To Produce Isolated Ultrabright Attosecond X-Ray Pulses}}, -volume = {14}, -year = {2011} -} -@article{VayAAC2010, -author = {Vay, J -. L and Geddes, C G R and Benedetti, C and Bruhwiler, D L and Cormier-Michel, E and Cowan, B M and Cary, J R and Grote, D P}, -doi = {10.1063/1.3520322}, -journal = {Aip Conference Proceedings}, -keywords = {[978-0-7354-0853-1/10/{\$}30.00]}, -pages = {244--249}, -title = {{Modeling Laser Wakefield Accelerators In A Lorentz Boosted Frame}}, -volume = {1299}, -year = {2010} -} -@article{Vayprl07, -author = {Vay, J.-L.}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -number = {13}, -pages = {130405/1--4}, -title = {{Noninvariance Of Space- And Time-Scale Ranges Under A Lorentz Transformation And The Implications For The Study Of Relativistic Interactions}}, -volume = {98}, -year = {2007} -} -@article{VayJCP2013, -abstract = {Pseudo-spectral electromagnetic solvers (i.e. representing the fields in Fourier space) have extraordinary precision. In particular, Haber et al. presented in 1973 a pseudo-spectral solver that integrates analytically the solution over a finite time step, under the usual assumption that the source is constant over that time step. Yet, pseudo-spectral solvers have not been widely used, due in part to the difficulty for efficient parallelization owing to global communications associated with global FFTs on the entire computational domains.A method for the parallelization of electromagnetic pseudo-spectral solvers is proposed and tested on single electromagnetic pulses, and on Particle-In-Cell simulations of the wakefield formation in a laser plasma accelerator.The method takes advantage of the properties of the Discrete Fourier Transform, the linearity of Maxwell's equations and the finite speed of light for limiting the communications of data within guard regions between neighboring computational domains.Although this requires a small approximation, test results show that no significant error is made on the test cases that have been presented.The proposed method opens the way to solvers combining the favorable parallel scaling of standard finite-difference methods with the accuracy advantages of pseudo-spectral methods. ?? 2013 Elsevier Inc.}, -author = {Vay, Jean Luc and Haber, Irving and Godfrey, Brendan B.}, -journal = {Journal of Computational Physics}, -keywords = {Domain decomposition,Electromagnetic,FFT,Fast fourier transform,Parallel,Particle-In-Cell,Spectral}, -pages = {260--268}, -title = {{A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas}}, -volume = {243}, -year = {2013} -} -@article{Kaganovichpop2004, -author = {Kaganovich, Id and Startsev, Ea and Davidson, Rc}, -doi = {10.1063/1.1758945}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {jul}, -number = {7}, -pages = {3546--3552}, -title = {{Nonlinear Plasma Waves Excitation By Intense Ion Beams In Background Plasma}}, -volume = {11}, -year = {2004} -} -@article{Cohenpop2005, -annote = {46Th Annual Meeting Of The Division Of Plasma Physics Of The American-Physical-Society, Savannah, Ga, Nov 15-19, 2004}, -author = {Cohen, Rh and Friedman, A and Covo, Mk and Lund, Sm and Molvik, Aw and Bieniosek, Fm and Seidl, Pa and Vay, Jl and Stoltz, P and Veitzer, S}, -doi = {10.1063/1.1882292}, -institution = {Amer Phys Soc}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -title = {{Simulating Electron Clouds In Heavy-Ion Accelerators}}, -volume = {12}, -year = {2005} -} -@article{Vayfed1996, -annote = {7Th International Symposium On Heavy Ion Inertial Fusion, Princeton Plasma Phys Lab, Princeton, Nj, Sep 06-09, 1995}, -author = {Vay, Jl and Deutsch, C}, -doi = {10.1016/S0920-3796(96)00502-9}, -issn = {0920-3796}, -journal = {Fusion Engineering And Design}, -month = {nov}, -pages = {467--476}, -title = {{A Three-Dimensional Electromagnetic Particle-In-Cell Code To Simulate Heavy Ion Beam Propagation In The Reaction Chamber}}, -volume = {32-33}, -year = {1996} -} -@article{Fengjcp09, -author = {Feng, B and Huang, C and Decyk, V and Mori, W B and Muggli, P and Katsouleas, T}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -number = {15}, -pages = {5340--5348}, -title = {{Enhancing Parallel Quasi-Static Particle-In-Cell Simulations With A Pipelining Algorithm}}, -volume = {228}, -year = {2009} -} -@article{BESAC2013, -abstract = {Presents a discussion regarding Montana Snow, the image on the cover of the present issue of American Psychologist, painted by John Axton. The discussion touches upon the artist's creative process, both in general and as it pertains to Montana Snow specifically. (PsycINFO Database Record (c) 2013 APA, all rights reserved).}, -author = {BESAC}, -doi = {10.1037/a0034573}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/582d1799e353fccb9458ff0ccb827db4b01d7308.pdf:pdf}, -issn = {1935-990X}, -journal = {The American psychologist}, -number = {7}, -pages = {601}, -pmid = {24128321}, -title = {{Directing Matter and Energy: Five Challenges for Science and the Imagination A}}, -url = {http://www.ncbi.nlm.nih.gov/pubmed/24130106}, -volume = {68}, -year = {2013} -} -@article{Prostprstab2005, -author = {Prost, Lr and Seidl, Pa and Bieniosek, Fm and Celata, Cm and Faltens, A and Baca, D and Henestroza, E and Kwan, Jw and Leitner, M and Waldron, Wl and Cohen, R and Friedman, A and Grote, D and Lund, Sm and Molvik, Aw and Morse, E}, -doi = {10.1103/Physrevstab.8.020101}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {feb}, -number = {2}, -title = {{High Current Transport Experiment For Heavy Ion Inertial Fusion}}, -volume = {8}, -year = {2005} -} -@inproceedings{Bassetti_Erskine, -author = {Bassetti, M and Erskine, G A}, -booktitle = {Cern Report No. Cern-Isrth/80-06}, -title = {{Closed Expression For The Electrical Field Of A Two-Dimensional Gaussian Charge}}, -year = {1980} -} -@inproceedings{Godfrey2013PPPS, -author = {Godfrey, B.{\~{}}B. and Haber, I and Vay, J.-L.}, -booktitle = {Proc. IEEE Pulsed Power and Plasma Science Conference}, -number = {4A-6}, -title = {{Numerical Stability of the Pseudo-Spectral EM PIC Algorithm}}, -year = {2013} -} -@article{Coleieee1997, -author = {Cole, Jb}, -issn = {0018-9480}, -journal = {Ieee Transactions On Microwave Theory And Techniques}, -month = {jun}, -number = {6}, -pages = {991--996}, -title = {{A High-Accuracy Realization Of The Yee Algorithm Using Non-Standard Finite Differences}}, -volume = {45}, -year = {1997} -} -@article{Vaypop04, -author = {Vay, J.-L. and Colella, P and Kwan, J W and Mccorquodale, P and Serafini, D B and Friedman, A and Grote, D P and Westenskow, G and Adam, J.-C. and Heron, A and Haber, I}, -doi = {10.1063/1.1689669}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -pages = {2928--2934}, -title = {{Application Of Adaptive Mesh Refinement To Particle-In-Cell Simulations Of Plasmas And Beams}}, -volume = {11}, -year = {2004} -} -@article{Friedmanjcp90, -author = {Friedman, A}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {oct}, -number = {2}, -pages = {292--312}, -title = {{A 2Nd-Order Implicit Particle Mover With Adjustable Damping}}, -volume = {90}, -year = {1990} -} -@article{Rittershoferpop2010, -author = {Rittershofer, W and Schroeder, C B and Esarey, E and Gruner, F J and Leemans, W P}, -doi = {10.1063/1.3430638}, -journal = {Physics Of Plasmas}, -month = {jun}, -number = {6}, -pages = {63104}, -title = {{Tapered Plasma Channels To Phase-Lock Accelerating And Focusing Forces In Laser-Plasma Accelerators}}, -volume = {17}, -year = {2010} -} -@article{Vorpal, -author = {Nieter, C and Cary, J R}, -journal = {J. Comput. Phys.}, -pages = {448--472}, -title = {{No Title}}, -volume = {196}, -year = {2004} -} -@article{LiARXIV2016, -abstract = {In this paper we present a customized finite-difference-time-domain (FDTD) Maxwell solver for the particle-in-cell (PIC) algorithm. The solver is customized to effectively eliminate the numerical Cerenkov instability (NCI) which arises when a plasma (neutral or non-neutral) relativistically drifts on a grid when using the PIC algorithm. We control the EM dispersion curve in the direction of the plasma drift of a FDTD Maxwell solver by using a customized higher order finite difference operator for the spatial derivative along the direction of the drift ({\$}\backslashhat 1{\$} direction). We show that this eliminates the main NCI modes with moderate {\$}\backslashvert k{\_}1 \backslashvert{\$}, while keeps additional main NCI modes well outside the range of physical interest with higher {\$}\backslashvert k{\_}1 \backslashvert{\$}. These main NCI modes can be easily filtered out along with first spatial aliasing NCI modes which are also at the edge of the fundamental Brillouin zone. The customized solver has the possible advantage of improved parallel scalability because it can be easily partitioned along {\$}\backslashhat 1{\$} which typically has many more cells than other directions for the problems of interest. We show that FFTs can be performed locally to current on each partition to filter out the main and first spatial aliasing NCI modes, and to correct the current so that it satisfies the continuity equation for the customized spatial derivative. This ensures that Gauss' Law is satisfied. We present simulation examples of one relativistically drifting plasmas, of two colliding relativistically drifting plasmas, and of nonlinear laser wakefield acceleration (LWFA) in a Lorentz boosted frame that show no evidence of the NCI can be observed when using this customized Maxwell solver together with its NCI elimination scheme.}, -archivePrefix = {arXiv}, -arxivId = {1605.01496}, -author = {Li, Fei and Yu, Peicheng and Xu, Xinlu and Fiuza, Frederico and Decyk, Viktor K. and Dalichaouch, Thamine and Davidson, Asher and Tableman, Adam and An, Weiming and Tsung, Frank S. and Fonseca, Ricardo A. and Lu, Wei and Mori, Warren B.}, -eprint = {1605.01496}, -month = {may}, -title = {{Controlling the Numerical Cerenkov Instability in PIC simulations using a customized finite difference Maxwell solver and a local FFT based current correction}}, -url = {http://arxiv.org/abs/1605.01496}, -year = {2016} -} -@article{XuJCP2013, -author = {Xu, Xinlu and Yu, Peicheng and Martins, Samual F and Tsung, Frank S and Decyk, Viktor K and Vieira, Jorge and Fonseca, Ricardo A and Lu, Wei and Silva, Luis O and Mori, Warren B}, -doi = {http://dx.doi.org/10.1016/j.cpc.2013.07.003}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -keywords = {Numerical Cherenkov radiation}, -number = {11}, -pages = {2503--2514}, -title = {{Numerical instability due to relativistic plasma drift in EM-PIC simulations}}, -url = {http://www.sciencedirect.com/science/article/pii/S0010465513002312}, -volume = {184}, -year = {2013} -} -@article{Cormierpre08, -author = {Cormier-Michel, Estelle and Shadwick, B A and Geddes, C G R and Esarey, E and Schroeder, C B and Leemans, W P}, -doi = {10.1103/Physreve.78.016404}, -issn = {1539-3755}, -journal = {Physical Review E}, -month = {jul}, -number = {1, Part 2}, -title = {{Unphysical Kinetic Effects In Particle-In-Cell Modeling Of Laser Wakefield Accelerators}}, -volume = {78}, -year = {2008} -} -@article{Berengerjcp96, -author = {Berenger, Jp}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {sep}, -number = {2}, -pages = {363--379}, -title = {{Three-Dimensional Perfectly Matched Layer For The Absorption Of Electromagnetic Waves}}, -volume = {127}, -year = {1996} -} -@article{Kalmykovprl09, -author = {{Kalmykov $\backslash$It Et Al.}, S}, -journal = {Phys. Rev. Lett.}, -pages = {135004}, -title = {{No Title}}, -volume = {103}, -year = {2009} -} -@article{Geddes2015, -abstract = {Near-monoenergetic photon sources at MeV energies offer improved sensitivity at greatly reduced dose for active interrogation, and new capabilities in treaty verification, nondestructive assay of spent nuclear fuel and emergency response. Thomson (also referred to as Compton) scattering sources are an established method to produce appropriate photon beams. Applications are however restricted by the size of the required high-energy electron linac, scattering (photon production) system, and shielding for disposal of the high energy electron beam. Laser-plasma accelerators (LPAs) produce GeV electron beams in centimeters, using the plasma wave driven by the radiation pressure of an intense laser. Recent LPA experiments are presented which have greatly improved beam quality and efficiency, rendering them appropriate for compact high-quality photon sources based on Thomson scattering. Designs for MeV photon sources utilizing the unique properties of LPAs are presented. It is shown that control of the scattering laser, including plasma guiding, can increase photon production efficiency. This reduces scattering laser size and/or electron beam current requirements to scale compatible with the LPA. Lastly, the plasma structure can decelerate the electron beam after photon production, reducing the size of shielding required for beam disposal. Together, these techniques provide a path to a compact photon source system.}, -author = {Geddes, Cameron G R and Rykovanov, Sergey and Matlis, Nicholas H. and Steinke, Sven and Vay, Jean Luc and Esarey, Eric H. and Ludewigt, Bernhard and Nakamura, Kei and Quiter, Brian J. and Schroeder, Carl B. and Toth, Csaba and Leemans, Wim P.}, -journal = {Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms}, -keywords = {Active interrogation,Homeland security,Laser plasma accelerator,Monoenergetic photon source,Nonproliferation}, -pages = {116--121}, -publisher = {Elsevier}, -title = {{Compact quasi-monoenergetic photon sources from laser-plasma accelerators for nuclear detection and characterization}}, -volume = {350}, -year = {2015} -} -@article{Antonsenprl1992, -author = {Antonsen, T M and Mora, P}, -doi = {10.1103/Physrevlett.69.2204}, -journal = {Physical Review Letters}, -month = {oct}, -number = {15}, -pages = {2204--2207}, -title = {{Self-Focusing And Raman-Scattering Of Laser-Pulses In Tenuous Plasmas}}, -volume = {69}, -year = {1992} -} -@article{GodfreyCPC2015, -abstract = {The family of generalized Pseudo-Spectral Time Domain (including the Pseudo-Spectral Analytical Time Domain) Particle-in-Cell algorithms offers substantial versatility for simulating particle beams and plasmas, and well written codes using these algorithms run reasonably fast. When simulating relativistic beams and streaming plasmas in multiple dimensions, they are, however, subject to the numerical Cherenkov instability. Previous studies have shown that instability growth rates can be reduced substantially by modifying slightly the transverse fields as seen by the streaming particles. Here, we offer an approach which completely eliminates the fundamental mode of the numerical Cherenkov instability while minimizing the transverse field corrections. The procedure, numerically computed residual growth rates (from weaker, higher order instability aliases), and comparisons with simulations using the code Warp are presented. In some instances, there are no numerical instabilities whatsoever, at least in the linear regime.}, -author = {Godfrey, Brendan B. and Vay, Jean Luc}, -journal = {Computer Physics Communications}, -keywords = {Numerical stability,Particle-in-cell,Pseudo-Spectral Time-Domain,Relativistic beam}, -pages = {221--225}, -publisher = {Elsevier}, -title = {{Improved numerical Cherenkov instability suppression in the generalized PSTD PIC algorithm}}, -volume = {196}, -year = {2015} -} -@article{VayCSD12, -author = {Vay, J.-L. and Grote, D P and Cohen, R H and Friedman, A}, -issn = {1749-4680}, -journal = {Computational Science and Discovery}, -number = {1}, -pages = {014019 (20 pp.)}, -title = {{Novel methods in the particle-in-cell accelerator code-framework warp}}, -type = {Journal Paper}, -volume = {5}, -year = {2012} -} -@article{Esirkepovcpc01, -author = {Esirkepov, Tz}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -month = {apr}, -number = {2}, -pages = {144--153}, -title = {{Exact Charge Conservation Scheme For Particle-In-Cell Simulation With An Arbitrary Form-Factor}}, -volume = {135}, -year = {2001} -} -@article{Martinspop10, -author = {Martins, S F and Fonseca, R A and Vieira, J and Silva, L O and Lu, W and Mori, W B}, -doi = {10.1063/1.3358139}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -pages = {56705}, -title = {{Modeling Laser Wakefield Accelerator Experiments With Ultrafast Particle-In-Cell Simulations In Boosted Frames}}, -volume = {17}, -year = {2010} -} -@inproceedings{Habericnsp73, -address = {Berkeley, Ca}, -author = {Haber, I and Lee, R and Klein, Hh and Boris, Jp}, -booktitle = {Proc. Sixth Conf. Num. Sim. Plasmas}, -pages = {46--48}, -title = {{Advances In Electromagnetic Simulation Techniques}}, -year = {1973} -} -@article{Martinscpc10, -author = {Martins, Samuel F and Fonseca, Ricardo A and Silva, Luis O and Lu, Wei and Mori, Warren B}, -doi = {10.1016/J.Cpc.2009.12.023}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -month = {may}, -number = {5}, -pages = {869--875}, -title = {{Numerical Simulations Of Laser Wakefield Accelerators In Optimal Lorentz Frames}}, -volume = {181}, -year = {2010} -} -@article{Vaycpc04, -author = {Vay, J.-L. and Adam, J.-C. and Heron, A}, -doi = {10.1016/J.Cpc.2004.06.026}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -month = {dec}, -number = {1-3}, -pages = {171--177}, -title = {{Asymmetric Pml For The Absorption Of Waves. Application To Mesh Refinement In Electromagnetic Particle-In-Cell Plasma Simulations}}, -volume = {164}, -year = {2004} -} -@article{GodfreyJCP2014_PSATD, -abstract = {The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC) algorithm solves the vacuum Maxwell's equations exactly, has no Courant time-step limit (as conventionally defined), and offers substantial flexibility in plasma and particle beam simulations. It is, however, not free of the usual numerical instabilities, including the numerical Cherenkov instability, when applied to relativistic beam simulations. This paper derives and solves the numerical dispersion relation for the PSATD algorithm and compares the results with corresponding behavior of the more conventional pseudo-spectral time-domain (PSTD) and finite difference time-domain (FDTD) algorithms. In general, PSATD offers superior stability properties over a reasonable range of time steps. More importantly, one version of the PSATD algorithm, when combined with digital filtering, is almost completely free of the numerical Cherenkov instability for time steps (scaled to the speed of light) comparable to or smaller than the axial cell size. ?? 2013 Elsevier Inc.}, -author = {Godfrey, Brendan B. and Vay, Jean Luc and Haber, Irving}, -journal = {Journal of Computational Physics}, -keywords = {Numerical stability,Particle-in-cell,Pseudo-spectral,Relativistic beam}, -pages = {689--704}, -title = {{Numerical stability analysis of the pseudo-spectral analytical time-domain PIC algorithm}}, -volume = {258}, -year = {2014} -} -@article{PruetJAP06, -author = {Pruet, J and McNabb, D P and Hagmann, C A and Hartemann, F V and Barty, C P J}, -doi = {10.1063/1.2202005}, -issn = {0021-8979}, -journal = {JOURNAL OF APPLIED PHYSICS}, -month = {jun}, -number = {12}, -title = {{Detecting clandestine material with nuclear resonance fluorescence}}, -volume = {99}, -year = {2006} -} -@article{YuCPC2015-Circ, -abstract = {A hybrid Maxwell solver for fully relativistic and electromagnetic (EM) particle-in-cell (PIC) codes is described. In this solver, the EM fields are solved in k space by performing an FFT in one direction, while using finite difference operators in the other direction(s). This solver eliminates the numerical Cerenkov radiation for particles moving in the preferred direction. Moreover, the numerical Cerenkov instability (NCI) induced by the relativistically drifting plasma and beam can be eliminated using this hybrid solver by applying strategies that are similar to those recently developed for pure FFT solvers. A current correction is applied for the charge conserving current deposit to ensure that Gauss's Law is satisfied. A theoretical analysis of the dispersion properties in vacuum and in a drifting plasma for the hybrid solver is presented, and compared with PIC simulations with good agreement obtained. This hybrid solver is applied to both 2D and 3D Cartesian and quasi-3D (in which the fields and current are decomposed into azimuthal harmonics) geometries. Illustrative results for laser wakefield accelerator simulation in a Lorentz boosted frame using the hybrid solver in the 2D Cartesian geometry are presented, and compared against results from 2D UPIC-EMMA simulation which uses a pure spectral Maxwell solver, and from OSIRIS 2D lab frame simulation using the standard Yee solver. Very good agreement is obtained which demonstrates the feasibility of using the hybrid solver for high fidelity simulation of relativistically drifting plasma with no evidence of the numerical Cerenkov instability. (C) 2015 Elsevier B.V. All rights reserved.}, -author = {Yu, Peicheng and Xu, Xinlu and Tableman, Adam and Decyk, Viktor K. and Tsung, Frank S. and Fiuza, Frederico and Davidson, Asher and Vieira, Jorge and Fonseca, Ricardo A. and Lu, Wei and Silva, Luis O. and Mori, Warren B.}, -doi = {10.1016/j.cpc.2015.08.026}, -issn = {00104655}, -journal = {Computer Physics Communications}, -keywords = {ALGORITHM,CODES,Hybrid Maxwell solver,IN-CELL SIMULATION,LASER WAKEFIELD ACCELERATORS,LORENTZ-BOOSTED FRAME,Numerical Cerenkov instability,OSIRIS,PARTICLE SIMULATION,PIC SIMULATIONS,PIC simulation,PLASMAS,Quasi-3D algorithm,Relativistic plasma drift,STABILITY}, -month = {dec}, -pages = {144--152}, -publisher = {ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS}, -title = {{Mitigation of numerical Cerenkov radiation and instability using a hybrid finite difference-FFT Maxwell solver and a local charge conserving current deposit}}, -url = {https://apps.webofknowledge.com/full{\_}record.do?product=UA{\&}search{\_}mode=GeneralSearch{\&}qid=2{\&}SID=1CanLFIHrQ5v8O7cxqV{\&}page=1{\&}doc=2}, -volume = {197}, -year = {2015} -} -@article{Berengerjcp94, -author = {Berenger, Jp}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {oct}, -number = {2}, -pages = {185--200}, -title = {{A Perfectly Matched Layer For The Absorption Of Electromagnetic-Waves}}, -volume = {114}, -year = {1994} -} -@article{Rubel2016, -author = {Rubel, Oliver and Loring, Burlen and Vay, Jean-Luc and Grote, David P. and Lehe, Remi and Bulanov, Stepan and Vincenti, Henri and Bethel, E. Wes}, -journal = {IEEE Computer Graphics and Applications}, -number = {Scientific Visualization}, -pages = {22--35}, -title = {{In situ Visualization and Analysis of Ion Accelerator Simulations using Warp and VisIt}}, -volume = {36}, -year = {2016} -} -@article{Geddesnature04, -author = {Geddes, Cgr and Toth, C and {Van Tilborg}, J and Esarey, E and Schroeder, Cb and Bruhwiler, D and Nieter, C and Cary, J and Leemans, Wp}, -doi = {10.1038/Nature02900}, -issn = {0028-0836}, -journal = {Nature}, -month = {sep}, -number = {7008}, -pages = {538--541}, -title = {{High-Quality Electron Beams From A Laser Wakefield Accelerator Using Plasma-Channel Guiding}}, -volume = {431}, -year = {2004} -} -@article{Munzjcp2000, -author = {Munz, Cd and Omnes, P and Schneider, R and Sonnendrucker, E and Voss, U}, -doi = {10.1006/Jcph.2000.6507}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {jul}, -number = {2}, -pages = {484--511}, -title = {{Divergence Correction Techniques For Maxwell Solvers Based On A Hyperbolic Model}}, -volume = {161}, -year = {2000} -} -@article{DavidsonJCP2015, -abstract = {For many plasma physics problems, three-dimensional and kinetic effects are very important. However, such simulations are very computationally intensive. Fortunately, there is a class of problems for which there is nearly azimuthal symmetry and the dominant three-dimensional physics is captured by the inclusion of only a few azimuthal harmonics. Recently, it was proposed [1] to model one such problem, laser wakefield acceleration, by expanding the fields and currents in azimuthal harmonics and truncating the expansion. The complex amplitudes of the fundamental and first harmonic for the fields were solved on an r–z grid and a procedure for calculating the complex current amplitudes for each particle based on its motion in Cartesian geometry was presented using a Marder's correction to maintain the validity of Gauss's law. In this paper, we describe an implementation of this algorithm into OSIRIS using a rigorous charge conserving current deposition method to maintain the validity of Gauss's law. We show that this algorithm is a hybrid method which uses a particles-in-cell description in r–z and a gridless description in ϕ. We include the ability to keep an arbitrary number of harmonics and higher order particle shapes. Examples for laser wakefield acceleration, plasma wakefield acceleration, and beam loading are also presented and directions for future work are discussed.}, -author = {Davidson, A. and Tableman, A. and An, W. and Tsung, F.S. and Lu, W. and Vieira, J. and Fonseca, R.A. and Silva, L.O. and Mori, W.B.}, -doi = {10.1016/j.jcp.2014.10.064}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Davidson et al. - 2015 - Implementation of a hybrid particle code with a PIC description in r–z and a gridless description in ϕ into.pdf:pdf}, -issn = {00219991}, -journal = {Journal of Computational Physics}, -pages = {1063--1077}, -title = {{Implementation of a hybrid particle code with a PIC description in r–z and a gridless description in ϕ into OSIRIS}}, -volume = {281}, -year = {2015} -} -@article{GodfreyJCP2014, -author = {Godfrey, Brendan B and Vay, Jean-Luc and Haber, Irving}, -doi = {http://dx.doi.org/10.1016/j.jcp.2013.10.053}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -keywords = {Numerical stability}, -number = {0}, -pages = {689--704}, -title = {{Numerical stability analysis of the pseudo-spectral analytical time-domain {\{}PIC{\}} algorithm}}, -url = {http://www.sciencedirect.com/science/article/pii/S0021999113007298}, -volume = {258}, -year = {2014} -} -@article{Godfrey2013, -abstract = {Rapidly growing numerical instabilities routinely occur in multidimensional particle-in-cell computer simulations of plasma-based particle accelerators, astrophysical phenomena, and relativistic charged particle beams. Reducing instability growth to acceptable levels has necessitated higher resolution grids, high-order field solvers, current filtering, etc. except for certain ratios of the time step to the axial cell size, for which numerical growth rates and saturation levels are reduced substantially. This paper derives and solves the cold beam dispersion relation for numerical instabilities in multidimensional, relativistic, electromagnetic particle-in-cell programs employing either the standard or the Cole-Karkkainnen finite difference field solver on a staggered mesh and the common Esirkepov current-gathering algorithm. Good overall agreement is achieved with previously reported results of the WARP code. In particular, the existence of select time steps for which instabilities are minimized is explained. Additionally, an alternative field interpolation algorithm is proposed for which instabilities are almost completely eliminated for a particular time step in ultra-relativistic simulations. ?? 2013 Elsevier Inc..}, -author = {Godfrey, Brendan B. and Vay, Jean Luc}, -journal = {Journal of Computational Physics}, -keywords = {Esirkepov algorithm,Numerical stability,Particle-in-cell,Relativistic beam}, -pages = {33--46}, -title = {{Numerical stability of relativistic beam multidimensional PIC simulations employing the Esirkepov algorithm}}, -volume = {248}, -year = {2013} -} -@article{Gilsonpop2010, -annote = {51St Annual Meeting Of The Division-Of-Plasma-Physics Of The American-Physics-Society, Atlanta, Ga, Nov 02-06, 2009}, -author = {Gilson, Erik P and Davidson, Ronald C and Dorf, Mikhail and Efthimion, Philip C and Majeski, Richard and Chung, Moses and Gutierrez, Michael S and Kabcenell, Aaron N}, -doi = {10.1063/1.3354109}, -institution = {Amer Phys Soc, Div Plasma Phys}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -title = {{Studies Of Emittance Growth And Halo Particle Production In Intense Charged Particle Beams Using The Paul Trap Simulator Experiment}}, -volume = {17}, -year = {2010} -} -@article{Cowanjcp11, -author = {Cowan, Benjamin M and Bruhwiler, David L and Cormier-Michel, Estelle and Esarey, Eric and Geddes, Cameron G R and Messmer, Peter and Paul, Kevin M}, -doi = {Doi: 10.1016/J.Jcp.2010.09.009}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -keywords = {Plasma Accelerator}, -number = {1}, -pages = {61--86}, -title = {{Characteristics Of An Envelope Model For Laser-Plasma Accelerator Simulation}}, -volume = {230}, -year = {2011} -} -@inproceedings{Geddesjp08, -author = {Geddes, C G R and Bruhwiler, D L and Cary, J R and Mori, W B and Vay, J.-L. and Martins, S F and Katsouleas, T and Cormier-Michel, E and Fawley, W M and Huang, C and Wang, X and Cowan, B and Decyk, V K and Esarey, E and Fonseca, R A and Lu, W and Messmer, P and Mullowney, P and Nakamura, K and Paul, K and Plateau, G R and Schroeder, C B and Silva, L O and Toth, C and Tsung, F S and Tzoufras, M and Antonsen, T and Vieira, J and Leemans, W P}, -booktitle = {Journal of Physics: Conference Series}, -issn = {1742-6596}, -pages = {012002 (11 Pp.)}, -title = {{Computational Studies And Optimization Of Wakefield Accelerators}}, -volume = {125}, -year = {2008} -} -@article{Steinke2016, -author = {Steinke, S and van Tilborg, J and Benedetti, C and Geddes, C G R and Schroeder, C B and Daniels, J and Swanson, K K and Gonsalves, A J and Nakamura, K and Matlis, N H and Shaw, B H and Esarey, E and Leemans, W P}, -issn = {0028-0836}, -journal = {Nature}, -month = {feb}, -number = {7589}, -pages = {190--193}, -publisher = {Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.}, -title = {{Multistage coupling of independent laser-plasma accelerators}}, -url = {http://dx.doi.org/10.1038/nature16525 http://10.1038/nature16525}, -volume = {530}, -year = {2016} -} -@article{Benedettiaac2010, -author = {Benedetti, C and Schroeder, C B and Esarey, E and Geddes, C G R and Leemans, W P}, -doi = {10.1063/1.3520323}, -journal = {Aip Conference Proceedings}, -keywords = {[978-0-7354-0853-1/10/{\$}30.00]}, -pages = {250--255}, -title = {{Efficient Modeling Of Laser-Plasma Accelerators With Inf{\&}Rno}}, -volume = {1299}, -year = {2010} -} -@article{Genonioppj2010, -author = {Genoni, T.{\~{}}C. and Clark, R.{\~{}}E. and {Van Welch}, D.{\~{}}R.}, -journal = {The Open Plasma Physics Journal}, -pages = {36}, -title = {{A Fast Implicit Algorithm For Highly Magnetized Charged Particle Motion}}, -volume = {3}, -year = {2010} -} -@article{LewisJCP1972, -author = {Lewis, H.Ralph}, -doi = {http://dx.doi.org/10.1016/0021-9991(72)90044-7}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -number = {3}, -pages = {400--419}, -title = {{Variational algorithms for numerical simulation of collisionless plasma with point particles including electromagnetic interactions}}, -url = {http://www.sciencedirect.com/science/article/pii/0021999172900447}, -volume = {10}, -year = {1972} -} -@article{Vay2002, -abstract = {The perfectly matched layer (PML) has become a standard for comparison in the techniques that have been developed to close the system of Maxwell equations (more generally, wave equations) when simulating an open system. The original Berenger PMLformulation relies on a split version of Maxwell equations, with numerical elec- tric and magnetic conductivities. We present here an extension of this formulation, which introduces counterparts of the electric and magnetic conductivities affecting the term which is spatially differentiated in the equations. The phase velocity along each direction is also multiplied by an additional coefficient. We show that under certain constraints on the additional numerical coefficients, this “medium” does not generate any reflection at any angle or any frequency and is thus a perfectly matched layer. Technically it is a superset of Berenger'sPMLto which it reduces for a specific set of parameters, and like it, it is anisotropic.However, unlike the PML, it introduces some asymmetry in the absorption rate and is therefore labeled an APML, for asym- metric perfectly matched layer. We present here the numerical considerations that have led us to introduce such a medium as well as its theory. Several finite-difference numerical implementations are derived (in one, two, and three dimensions) and the performance of the APML is contrasted with that of the PML in one and two di- mensions. Using plane wave analysis, we show that our APML implementations lead to higher absorption rates than the considered PML implementations. Although we have considered in this paper the finite-difference discretization of Maxwell-like equations only, the APML system of equations may be used with other discretiza- tion schemes, such as finite elements, and may be applied to other equations, for applications beyond electromagnetics.}, -author = {Vay, Jean-Luc}, -journal = {Journal of Computational Physics}, -keywords = {PML,a nonexclusive royalty-free license,abc,absorbing boundary condition,copyright covering this,electromagnetics,fdtd,finite-difference time domain,for governmental purposes,government,in and to the,is acknowledged,maxwell,paper,perfectly matched layer,pml,s,s right to retain,the u,wave equation}, -number = {2}, -pages = {367--399}, -title = {{Asymmetric Perfectly Matched Layer for the Absorption of Waves☆}}, -url = {http://linkinghub.elsevier.com/retrieve/pii/S0021999102971755}, -volume = {183}, -year = {2002} -} -@article{Manglesnature04, -author = {Mangles, Spd and Murphy, Cd and Najmudin, Z and Thomas, Agr and Collier, Jl and Dangor, Ae and Divall, Ej and Foster, Ps and Gallacher, Jg and Hooker, Cj and Jaroszynski, Da and Langley, Aj and Mori, Wb and Norreys, Pa and Tsung, Fs and Viskup, R and Walton, Br and Krushelnick, K}, -doi = {10.1038/Nature02939}, -issn = {0028-0836}, -journal = {Nature}, -month = {sep}, -number = {7008}, -pages = {535--538}, -title = {{Monoenergetic Beams Of Relativistic Electrons From Intense Laser-Plasma Interactions}}, -volume = {431}, -year = {2004} -} -@inproceedings{Schroederaac08, -author = {Schroeder, C B and Esarey, E and Geddes, C G R and Toth, C and Leemans, W P}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -pages = {208--214}, -title = {{Design Considerations For A Laser-Plasma Linear Collider}}, -volume = {1086}, -year = {2009} -} -@article{Vaypop98, -author = {Vay, J.-L. and Deutsch, C}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {apr}, -number = {4}, -pages = {1190--1197}, -title = {{Charge Compensated Ion Beam Propagation In A Reactor Sized Chamber}}, -volume = {5}, -year = {1998} -} -@article{Friedmanjcp1991, -author = {Friedman, A and Parker, Se and Ray, Sl and Birdsall, Ck}, -doi = {10.1016/0021-9991(91)90265-M}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {sep}, -number = {1}, -pages = {54--70}, -title = {{Multiscale Particle-In-Cell Plasma Simulation}}, -volume = {96}, -year = {1991} -} -@article{Liumotl1997, -author = {Liu, Qh}, -doi = {10.1002/(Sici)1098-2760(19970620)15:3<158::Aid-Mop11>3.3.Co;2-T}, -issn = {0895-2477}, -journal = {Microwave And Optical Technology Letters}, -month = {jun}, -number = {3}, -pages = {158--165}, -title = {{The PSTD Algorithm: A Time-Domain Method Requiring Only Two Cells Per Wavelength}}, -volume = {15}, -year = {1997} -} -@inproceedings{Karkicap06, -address = {Chamonix, France}, -author = {Karkkainen, M and Gjonaj, E and Lau, T and Weiland, T}, -booktitle = {Proc. Of International Computational Accelerator Physics Conference}, -pages = {35--40}, -title = {{Low-Dispersionwake Field Calculation Tools}}, -year = {2006} -} -@article{Schroederprstab10, -author = {Schroeder, C.{\~{}}B. and Esarey, E and Geddes, C.{\~{}}G.{\~{}}R. and Benedetti, C and Leemans, W.{\~{}}P.}, -journal = {Phys. Rev. Spec. Topics Accel. Beams.}, -pages = {101301}, -title = {{Physics Considerations For Laser-Plasma Linear Colliders}}, -volume = {13}, -year = {2010} -} -@article{Leemansphysicstoday10, -author = {Leemans, Wim and Esarey, Eric}, -issn = {0031-9228}, -journal = {Physics Today}, -month = {mar}, -number = {3}, -pages = {44--49}, -title = {{Laser-Driven Plasma-Wave Electron Accelerators}}, -volume = {62}, -year = {2009} -} -@article{Lehe2015, -abstract = {We propose a spectral Particle-In-Cell (PIC) algorithm that is based on the combination of a Hankel transform and a Fourier transform. For physical problems that have close-to-cylindrical symmetry, this algorithm can be much faster than full 3D PIC algorithms. In addition, unlike standard finite-difference PIC codes, the proposed algorithm is free of numerical dispersion. This algorithm is benchmarked in several situations that are of interest for laser-plasma interactions. These benchmarks show that it avoids a number of numerical artifacts, that would otherwise affect the physics in a standard PIC algorithm - including the zero-order numerical Cherenkov effect.}, -archivePrefix = {arXiv}, -arxivId = {1507.04790}, -author = {Lehe, Remi and Kirchen, Manuel and Andriyash, Igor a. and Godfrey, Brendan B. and Vay, Jean-Luc}, -eprint = {1507.04790}, -isbn = {5104866785}, -journal = {arXiv.org}, -keywords = {cylindrical geometry,hankel transform,particle-in-cell,pseudo-spectral}, -pages = {1507.04790v1}, -title = {{A spectral, quasi-cylindrical and dispersion-free Particle-In-Cell algorithm}}, -url = {http://arxiv.org/abs/1507.04790}, -volume = {physics.pl}, -year = {2015} -} -@article{VayJCP13, -author = {Vay, Jean-Luc and Haber, Irving and Godfrey, Brendan B}, -doi = {10.1016/j.jcp.2013.03.010}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {jun}, -pages = {260--268}, -title = {{A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas}}, -volume = {243}, -year = {2013} -} -@inproceedings{Martinspac09, -address = {Vancouver, Canada}, -annote = {Th4Gbc05}, -author = {{Martins $\backslash$It Et Al.}, S F}, -booktitle = {Proc. Particle Accelerator Conference}, -title = {{Boosted Frame Pic Simulations Of Lwfa: Towards The Energy Frontier}}, -year = {2009} -} -@article{Vayscidac09, -author = {Vay, J.-L. and Bruhwiler, D L and Geddes, C G R and Fawley, W M and Martins, S F and Cary, J R and Cormier-Michel, E and Cowan, B and Fonseca, R A and Furman, M A and Lu, W and Mori, W B and Silva, L O}, -issn = {1742-6596}, -journal = {Journal of Physics: Conference Series}, -number = {1}, -pages = {012006 (5 Pp.)}, -title = {{Simulating Relativistic Beam And Plasma Systems Using An Optimal Boosted Frame}}, -volume = {180}, -year = {2009} -} -@article{Rykovanov2014, -author = {Rykovanov, S.{\~{}}G.}, -journal = {in preparation}, -title = {{No Title}} -} -@article{Sprangleprl90, -author = {Sprangle, P and Esarey, E and Ting, A}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {apr}, -number = {17}, -pages = {2011--2014}, -title = {{Nonlinear-Theory Of Intense Laser-Plasma Interactions}}, -volume = {64}, -year = {1990} -} -@article{Molvikpop2007, -annote = {48Th Annual Meeting Of The Division Of Plasma Physics Of The Aps, Philadelphia, Pa, Jan 30-Nov 03, 2006}, -author = {Molvik, A W and Covo, M Kireeff and Cohen, R and Friedman, A and Lund, S M and Sharp, W and Vay, J-L. and Baca, D and Bieniosek, F and Leister, C and Seidl, P}, -doi = {10.1063/1.2436850}, -institution = {Aps, Div Plasma Phys}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -title = {{Quantitative Experiments With Electrons In A Positively Charged Beam}}, -volume = {14}, -year = {2007} -} -@article{QuiterJAP08, -author = {Quiter, B J and Prussin, S G and Pohl, B and Hall, J and Trebes, J and Stone, G and Descalle, M -A.}, -doi = {10.1063/1.2876028}, -issn = {0021-8979}, -journal = {JOURNAL OF APPLIED PHYSICS}, -month = {mar}, -number = {6}, -title = {{A method for high-resolution x-ray imaging of intermodal cargo containers for fissionable materials}}, -volume = {103}, -year = {2008} -} -@book{Taflove2000, -address = {Norwood}, -author = {Taflove and Hagness}, -edition = {2Nd}, -publisher = {Ma: Artech House}, -title = {{Computational Electrodynamics: The Finite-Difference Time-Domain Method}}, -year = {2000} -} -@article{Schroederprl2011, -author = {Schroeder, C B and Benedetti, C and Esarey, E and Leemans, W P}, -doi = {10.1103/Physrevlett.106.135002}, -journal = {Physical Review Letters}, -month = {mar}, -number = {13}, -pages = {135002}, -title = {{Nonlinear Pulse Propagation And Phase Velocity Of Laser-Driven Plasma Waves}}, -volume = {106}, -year = {2011} -} -@article{Logannim2007, -annote = {Proceedings Of The 16Th International Symposium On Heavy Ion Inertial Fusion Hif 06}, -author = {Logan, B G and Bieniosek, F M and Celata, C M and Coleman, J and Greenway, W and Henestroza, E and Kwan, J W and Lee, E P and Leitner, M and Roy, P K and Seidl, P A and Vay, J.-L. and Waldron, W L and Yu, S S and Barnard, J J and Cohen, R H and Friedman, A and Grote, D P and Covo, M Kireeff and Molvik, A W and Lund, S M and Meier, W R and Sharp, W and Davidson, R C and Efthimion, P C and Gilson, E P and Grisham, L and Kaganovich, I D and Qin, H and Sefkow, A B and Startsev, E A and Welch, D and Olson, C}, -doi = {10.1016/J.Nima.2007.02.070}, -issn = {0168-9002}, -journal = {Nuclear Instruments And Methods In Physics Research Section A: Accelerators, Spectrometers, Detectors And Associated Equipment}, -keywords = {Heavy Ion Beams}, -number = {1--2}, -pages = {1--7}, -title = {{Recent Us Advances In Ion-Beam-Driven High Energy Density Physics And Heavy Ion Fusion}}, -url = {Http://www.sciencedirect.com/Science/Article/Pii/S0168900207002847}, -volume = {577}, -year = {2007} -} -@article{Yu2016, -abstract = {When modeling laser wakefield acceleration (LWFA) using the particle-in-cell (PIC) algorithm in a Lorentz boosted frame, the plasma is drifting relativistically at $\beta$bc towards the laser, which can lead to a computational speedup of ∼$\gamma$b2=(1−$\beta$b2)−1. Meanwhile, when LWFA is modeled in the quasi-3D geometry in which the electromagnetic fields and current are decomposed into a limited number of azimuthal harmonics, speedups are achieved by modeling three dimensional (3D) problems with the computational loads on the order of two dimensional r−z simulations. Here, we describe a method to combine the speed ups from the Lorentz boosted frame and quasi-3D algorithms. The key to the combination is the use of a hybrid Yee-FFT solver in the quasi-3D geometry that significantly mitigates the Numerical Cerenkov Instability (NCI) which inevitably arises in a Lorentz boosted frame due to the unphysical coupling of Langmuir modes and EM modes of the relativistically drifting plasma in these simulations. In addition, based on the space-time distribution of the LWFA data in the lab and boosted frame, we propose to use a moving window to follow the drifting plasma, instead of following the laser driver as in the LWFA lab frame simulation, in order to further reduce the computational loads. We describe the details of how the NCI is mitigated for the quasi-3D geometry, the setups for simulations which combine the Lorentz boosted frame, quasi-3D geometry, and the use of a moving window, and compare the results from these simulations against their corresponding lab frame cases. Good agreement is obtained among these sample simulations, particularly when there is no self-trapping, which demonstrates it is possible to combine the Lorentz boosted frame and the quasi-3D algorithms when modeling LWFA. We also discusse the preliminary speedups achieved in these sample simulations.}, -author = {Yu, Peicheng and Xu, Xinlu and Davidson, Asher and Tableman, Adam and Dalichaouch, Thamine and Li, Fei and Meyers, Michael D. and An, Weiming and Tsung, Frank S. and Decyk, Viktor K. and Fiuza, Frederico and Vieira, Jorge and Fonseca, Ricardo A. and Lu, Wei and Silva, Luis O. and Mori, Warren B.}, -doi = {10.1016/j.jcp.2016.04.014}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Yu et al. - 2016 - Enabling Lorentz boosted frame particle-in-cell simulations of laser wakefield acceleration in quasi-3D geometry.pdf:pdf}, -issn = {00219991}, -journal = {Journal of Computational Physics}, -title = {{Enabling Lorentz boosted frame particle-in-cell simulations of laser wakefield acceleration in quasi-3D geometry}}, -year = {2016} -} -@article{Borisjcp73, -address = {525 B St, Ste 1900, San Diego, Ca 92101-4495}, -author = {Boris, Jp and Lee, R}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -number = {1}, -pages = {131--136}, -publisher = {Academic Press Inc Jnl-Comp Subscriptions}, -title = {{Nonphysical Self Forces In Some Electromagnetic Plasma-Simulation Algorithms}}, -type = {Note}, -volume = {12}, -year = {1973} -} -@inproceedings{BorisICNSP70, -address = {Naval Res. Lab., Wash., D. C.}, -author = {Boris, Jp}, -booktitle = {Proc. Fourth Conf. Num. Sim. Plasmas}, -pages = {3--67}, -title = {{Relativistic Plasma Simulation-Optimization of a Hybrid Code}}, -year = {1970} -} -@article{Vaya, -author = {Vay, Jean-Luc and Godfrey, Brendan and Haber, Irving and Lehe, R{\'{e}}mi and Vincenti, Henri}, -title = {{In preparation}} -} -@article{Vayjcp01, -author = {Vay, J.-L.}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {feb}, -number = {1}, -pages = {72--98}, -title = {{An Extended Fdtd Scheme For The Wave Equation: Application To Multiscale Electromagnetic Simulation}}, -volume = {167}, -year = {2001} -} -@inproceedings{Posinstlbl2002, -address = {Berkeley, Ca, Usa}, -author = {Furman, M and Pivi, M T F}, -booktitle = {Lbnl-49771/Cbp Note-4151}, -title = {{No Title}}, -year = {2002} -} -@misc{Kirchen2016, -author = {Kirchen, Manuel and Lehe, R{\'{e}}mi}, -title = {{Accelerating a Spectral Algorithm for Plasma Physics with Python/Numba on GPU}}, -url = {http://on-demand.gputechconf.com/gtc/2016/presentation/s6353-manuel-kirchen-spectral-algorithm-plasma-physics.pdf}, -year = {2016} -} -@article{Leemansnature06, -author = {Leemans, W P and Nagler, B and Gonsalves, A J and Toth, Cs. and Nakamura, K and Geddes, C G R and Esarey, E and Schroeder, C B and Hooker, S M}, -doi = {10.1038/Nphys418}, -issn = {1745-2473}, -journal = {Nature Physics}, -month = {oct}, -number = {10}, -pages = {696--699}, -title = {{Gev Electron Beams From A Centimetre-Scale Accelerator}}, -volume = {2}, -year = {2006} -} -@article{ChenPRSTAB13, -author = {Chen, M and Esarey, E and Geddes, C G R and Schroeder, C B and Plateau, G R and Bulanov, S S and Rykovanov, S and Leemans, W P}, -doi = {10.1103/PhysRevSTAB.16.030701}, -issn = {1098-4402}, -journal = {PHYSICAL REVIEW SPECIAL TOPICS-ACCELERATORS AND BEAMS}, -month = {mar}, -number = {3}, -title = {{Modeling classical and quantum radiation from laser-plasma accelerators}}, -volume = {16}, -year = {2013} -} -@article{Hipace, -author = {Mehrling, T and Benedetti, C and Schroeder, C B and Osterhoff, J}, -doi = {10.1088/0741-3335/56/8/084012}, -issn = {0741-3335}, -journal = {Plasma Physics and Controlled Fusion}, -month = {aug}, -number = {8}, -pages = {084012}, -publisher = {IOP Publishing}, -title = {{HiPACE: a quasi-static particle-in-cell code}}, -url = {http://stacks.iop.org/0741-3335/56/i=8/a=084012?key=crossref.70bffad7eee174a2aacc367aee0a7216}, -volume = {56}, -year = {2014} -} -@article{Morsenielson1971, -author = {Morse, Rl and Nielson, Cw}, -doi = {10.1063/1.1693518}, -issn = {1070-6631}, -journal = {Phys. Fluids}, -number = {4}, -pages = {830--{\&}}, -title = {{Numerical Simulation Of Weibel Instability In One And 2 Dimensions}}, -volume = {14}, -year = {1971} -} -@article{Vaydpf09, -author = {{Vay $\backslash$It Et Al.}, J.-L.}, -journal = {Arxiv:0909.5603}, -title = {{Speeding Up Simulations Of Relativistic Systems Using An Optimal Boosted Frame}}, -year = {2009} -} -@misc{Vay2014, -abstract = {Standard methods employed in relativistic electromagnetic Particle-In-Cell codes are reviewed, as well as novel techniques that were introduced recently. Advances in the analysis and mitigation of the numerical Cherenkov instability are also presented with comparison between analytical theory and numerical experiments. The algorithmic and numerical analytic advances are expanding the range of applicability of the method in the ultra-relativistic regime in particular, where the numerical Cherenkov instability is the strongest without corrective measures.}, -author = {Vay, Jean Luc and Godfrey, Brendan B.}, -booktitle = {Comptes Rendus - Mecanique}, -keywords = {Numerical instability,Particle-In-Cell,Plasma simulation,Special relativity}, -number = {10-11}, -pages = {610--618}, -publisher = {Elsevier Masson SAS}, -title = {{Modeling of relativistic plasmas with the Particle-In-Cell method}}, -volume = {342}, -year = {2014} -} -@article{Lehearxiv2015, -author = {Lehe, R and Kirchen, M and Andriyash, I.{\~{}}A. and Godfrey, B.{\~{}}B. and Vay, J.-L.}, -journal = {arXiv:1507.04790}, -title = {{A spectral, quasi-cylindrical and dispersion-free Particle-In-Cell algorithm}}, -year = {2015} -} -@article{Friedman2014, -abstract = {The Warp code (and its framework of associated tools) was initially developed for particle-in-cell simulations of space-charge-dominated ion beams in accelerators, for heavy-ion-driven inertial fusion energy, and related experiments. It has found a broad range of applications, including nonneutral plasmas in traps, stray electron clouds in accelerators, laser-based acceleration, and the focusing of ion beams produced when short-pulse lasers irradiate foil targets. We summarize novel methods used in Warp, including: time-stepping conducive to diagnosis and particle injection; an interactive Python-Fortran-C structure that enables scripted and interactive user steering of runs; a variety of geometries (3-D x, y, z; 2-D r, z; 2-D x, y); electrostatic and electromagnetic field solvers; a cut-cell representation for internal boundaries; the use of warped coordinates for bent beam lines; adaptive mesh refinement, including a capability for time-dependent space-charge-limited flow from curved surfaces; models for accelerator lattice elements (magnetic or electrostatic quadrupole lenses, accelerating gaps, etc.) at user-selectable levels of detail; models for particle interactions with gas and walls; moment/envelope models that support sophisticated particle loading; a drift-Lorentz mover for rapid tracking through regions of strong and weak magnetic field; a Lorentz-boosted frame formulation with a Lorentz-invariant modification of the Boris mover; an electromagnetic solver with tunable dispersion and stride-based digital filtering; and a pseudospectral electromagnetic solver. Warp has proven useful for a wide range of applications, described very briefly herein. It is available as an open-source code under a BSD license. This paper describes material presented during the Prof. Charles K. (Ned) Birdsall Memorial Session of the 2013 IEEE Pulsed Power and Plasma Science Conference. In addition to our overview of the computational methods used in Warp, we summarize a few asp- cts of Ned's contributions to plasma simulation and to the careers of those he mentored.}, -author = {Friedman, Alex and Cohen, Ronald H. and Grote, David P. and Lund, Steven M. and Sharp, William M. and Vay, Jean Luc and Haber, Irving and Kishek, Rami A.}, -journal = {IEEE Transactions on Plasma Science}, -keywords = {Algorithms,Maxwell,Ned Birdsall,computer,laser,numerical simulation,particle beam,particle-in-cell,plasma}, -number = {5}, -pages = {1321--1334}, -publisher = {Institute of Electrical and Electronics Engineers Inc.}, -title = {{Computational methods in the warp code framework for kinetic simulations of particle beams and plasmas}}, -volume = {42}, -year = {2014} -} -@article{Lcode, -abstract = {To study the long-term dynamics of ultrarelativistic particle beams (drivers) in a plasma wake-field accelerator, a two-dimensional hybrid code, LCODE, has been developed, in which the plasma is treated as an electron fluid, while the beam is treated as an ensemble of particles. With LCODE the evolution of the driver composed of several short particle bunches is studied. When properly positioned, the bunches are shown to achieve a radial equilibrium in which all driver particles are focused by the plasma, and the focusing force is balanced by driver transverse pressure. After establishment of the radial equilibrium, the driver loses much of its energy with little distortion in shape. Losing a major part of the initial energy, driver particles get defocused and lost. Due to this “self-cleaning,” the driver generates a good wake-field even after the particle loss of more than the half.}, -author = {Lotov, K. V.}, -doi = {10.1063/1.872765}, -issn = {1070664X}, -journal = {Physics of Plasmas}, -number = {3}, -pages = {785}, -publisher = {AIP Publishing}, -title = {{Simulation of ultrarelativistic beam dynamics in plasma wake-field accelerator}}, -url = {http://scitation.aip.org/content/aip/journal/pop/5/3/10.1063/1.872765}, -volume = {5}, -year = {1998} -} -@article{GodfreyJCP2014_2, -author = {Godfrey, Brendan B and Vay, Jean-Luc}, -doi = {http://dx.doi.org/10.1016/j.jcp.2014.02.022}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -keywords = {Numerical stability}, -number = {0}, -pages = {1--6}, -title = {{Suppressing the numerical Cherenkov instability in {\{}FDTD{\}} {\{}PIC{\}} codes}}, -url = {http://www.sciencedirect.com/science/article/pii/S0021999114001429}, -volume = {267}, -year = {2014} -} -@article{Turbowave, -abstract = {—A novel particle simulation code is described that self-consistently models certain classes of laser–plasma inter-actions without resolving the optical cycles of the laser. This is accomplished by separating the electromagnetic field into a laser component and a wake component. Although the wake component is treated as in a fully explicit particle-in-cell (PIC) code, the laser component is treated in the high-frequency limit, which allows the optical cycles to be averaged out. This leads to enormous reductions in computer time when the laser frequency is much greater than all other frequencies of interest. This work is an extension of the work of Mora and Antonsen, Jr. [1], [2], who derived the time-averaged equations coupling the laser with the particles and developed a code to solve these equations in the quasi-static limit. The code presented here is distinguished by the fact that it is useful when the plasma length is much less than the laser pulse length. Also, it is already parallelized and should be straightforward to extend to three dimensions.}, -author = {Gordon, Daniel F and Mori, W B and Antonsen, Thomas M}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Gordon, Mori, Antonsen - 2000 - A Ponderomotive Guiding Center Particle-in-Cell Code for Efficient Modeling of Laser–Plasma Interactio.pdf:pdf}, -journal = {IEEE TRANSACTIONS ON PLASMA SCIENCE}, -keywords = {Index Terms—Particle code,plasma simulation}, -number = {4}, -title = {{A Ponderomotive Guiding Center Particle-in-Cell Code for Efficient Modeling of Laser–Plasma Interactions}}, -volume = {28}, -year = {2000} -} -@article{Habernim2009, -annote = {17Th International Symposium On Heavy Ion Inertial Fusion, Tokyo, Japan, Aug 04-08, 2008}, -author = {Haber, I and Bernal, S and Beaudoin, B and Cornacchia, A and Feldman, D and Feldman, R B and Fiorito, R and Fiuza, K and Godlove, T F and Kishek, R A and O'shea, P G and Quinn, B and Papadopoulos, C and Reiser, M and Stratakis, D and Sutter, D and Thangaraj, J C T and Tian, K and Walter, M and Wu, C}, -doi = {10.1016/J.Nima.2009.03.220}, -institution = {Tokyo Inst Technol, Res Lab Nucl Reactors; Japan Soc Plasma Sci {\&} Nucl Fus Res; Particle Accelerator Soc Japan}, -issn = {0168-9002}, -journal = {Nuclear Instruments {\&} Methods In Physics Research Section A-Accelerators Spectrometers Detectors And Associated Equipment}, -month = {jul}, -number = {1-2}, -pages = {64--68}, -title = {{Scaled Electron Studies At The University Of Maryland}}, -volume = {606}, -year = {2009} -} -@article{Folegatijpcs2011, -author = {Folegati, Paola and Xu, Jia and Weber, Marc H and Lynn, Kelvin G}, -journal = {Journal of Physics: Conference Series}, -number = {1}, -pages = {12021}, -title = {{Positron Storage In Micro-Traps With Long Aspect Ratio: Results Of Computer Simulations}}, -url = {Http://stacks.iop.org/1742-6596/262/I=1/A=012021}, -volume = {262}, -year = {2011} -} -@article{YuCPC2015, -abstract = {When using an electromagnetic particle-in-cell (EM-PIC) code to simulate a relativistically drifting plasma, a violent numerical instability known as the numerical Cerenkov instability (NCI) occurs. The NCI is due to the unphysical coupling of electromagnetic waves on a grid to wave-particle resonances, including aliased resonances, i.e., omega + 2 pi mu/Delta t = (k(1) + 2 pi v(1)/Delta x(1))nu(0), where mu and v(1) refer to the time and space aliases and the plasma is drifting relativistically at velocity nu(0) in the (1) over cap -direction. We extend our previous work Xu et al. (2013) by recasting the numerical dispersion relation of a relativistically drifting plasma into a form which shows explicitly how the instability results from the coupling modes which are-purely-transverse electromagnetic (EM) modes-and purely longitudinal modes in-the rest frame of the plasma for each time and space aliasing. The dispersion relation for each mu and v(1) is the product of the dispersion relation of these two modes set equal to a coupling term that vanishes in the continuous limit. The new form of the numerical dispersion relation provides an accurate method of systematically calculating the growth rate and location of the mode in the fundamental Brillouin zone for any Maxwell solver for each mu, and v(1). We then focus on the spectral Maxwell solver and systematically discuss its NCI modes. We show that the second fastest growing NCI mode for the spectral solver corresponds to mu = v(1) = 0, that it has a growth rate approximately one order of magnitude smaller than the fastest growing mu = 0 and v(1) = 1 mode, and that its location in the k space fundamental Brillouin zone is sensitive to the grid size and time step. Based on these studies, strategies to systematically eliminate the NCI modes for a spectral solver are developed. We apply these strategies to both relativistic collisionless shock and LWFA simulations, and demonstrate that high-fidelity multi-dimensional simulations of drifting plasmas can be carried out with a spectral Maxwell solver with no evidence of numerical Cerenkov instability. (C) 2015 Elsevier B.V. All rights reserved.}, -author = {Yu, Peicheng and Xu, Xinlu and Decyk, Viktor K. and Fiuza, Frederico and Vieira, Jorge and Tsung, Frank S. and Fonseca, Ricardo A. and Lu, Wei and Silva, Luis O. and Mori, Warren B.}, -doi = {10.1016/j.cpc.2015.02.018}, -issn = {00104655}, -journal = {Computer Physics Communications}, -keywords = {ALGORITHM,LASER WAKEFIELD ACCELERATORS,LORENTZ-BOOSTED FRAME,Numerical Cerenkov instability,Numerical dispersion relation,PARTICLE SIMULATION,PLASMA,Particle-in-cell,Plasma simulation,Relativistic drifting plasma,SHOCKS,STABILITY,Spectral solver,WAVES}, -month = {jul}, -pages = {32--47}, -publisher = {ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS}, -title = {{Elimination of the numerical Cerenkov instability for spectral EM-PIC codes}}, -url = {https://apps.webofknowledge.com/full{\_}record.do?product=UA{\&}search{\_}mode=GeneralSearch{\&}qid=2{\&}SID=1CanLFIHrQ5v8O7cxqV{\&}page=1{\&}doc=3}, -volume = {192}, -year = {2015} -} -@article{PukhovJPP99, -author = {Pukhov, A}, -doi = {10.1017/S0022377899007515}, -issn = {0022-3778}, -journal = {Journal of Plasma Physics}, -month = {apr}, -number = {3}, -pages = {425--433}, -title = {{Three-dimensional electromagnetic relativistic particle-in-cell code VLPL (Virtual Laser Plasma Lab)}}, -volume = {61}, -year = {1999} -} -@article{Vincentiarxiv2015, -author = {Vincenti, H and Vay, J.-L.}, -journal = {arXiv:1507.05572}, -title = {{Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference Maxwell solver}}, -year = {2015} -} -@inproceedings{Vayipac10, -address = {Tokyo, Japan}, -annote = {Weobra02}, -author = {Vay, J.-L. and Byrd, J.{\~{}}M. and Furman, M.{\~{}}A. and Secondo, R and Venturini, M and Fox, J.{\~{}}D. and Rivetta, C.{\~{}}H. and H{\"{O}}fle, W}, -booktitle = {Proc. 1St International Particle Accelerator Conference}, -title = {{Simulation Of E-Cloud Driven Instability And Its Attenuation Using A Feedback System In The Cern Sps}}, -year = {2010} -} -@article{Cormierprstab2011, -author = {Cormier-Michel, E and Esarey, E and Geddes, C G R and Schroeder, C B and Paul, K and Mullowney, P J and Cary, J R and Leemans, W P}, -doi = {10.1103/Physrevstab.14.031303}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {mar}, -number = {3}, -pages = {31303}, -title = {{Control Of Focusing Fields In Laser-Plasma Accelerators Using Higher-Order Modes}}, -volume = {14}, -year = {2011} -} -@article{Lehe2016, -abstract = {We propose a spectral Particle-In-Cell (PIC) algorithm that is based on the combination of a Hankel transform and a Fourier transform. For physical problems that have close-to-cylindrical symmetry, this algorithm can be much faster than full 3D PIC algorithms. In addition, unlike standard finite-difference PIC codes, the proposed algorithm is free of spurious numerical dispersion, in vacuum. This algorithm is benchmarked in several situations that are of interest for laser–plasma interactions. These benchmarks show that it avoids a number of numerical artifacts, that would otherwise affect the physics in a standard PIC algorithm — including the zero-order numerical Cherenkov effect.}, -author = {Lehe, R{\'{e}}mi and Kirchen, Manuel and Andriyash, Igor A. and Godfrey, Brendan B. and Vay, Jean-Luc}, -doi = {10.1016/j.cpc.2016.02.007}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Lehe et al. - 2016 - A spectral, quasi-cylindrical and dispersion-free Particle-In-Cell algorithm.pdf:pdf}, -issn = {00104655}, -journal = {Computer Physics Communications}, -pages = {66--82}, -title = {{A spectral, quasi-cylindrical and dispersion-free Particle-In-Cell algorithm}}, -volume = {203}, -year = {2016} -} -@book{Birdsalllangdon, -author = {Birdsall, C K and Langdon, A B}, -isbn = {0 07 005371 5}, -pages = {Xxvi+479 Pp.}, -publisher = {Adam-Hilger}, -title = {{Plasma Physics Via Computer Simulation}}, -year = {1991} -} -@inproceedings{Grote2005, -abstract = {The Warp code, developed for heavy‐ion driven inertial fusion energy studies, is used to model high intensity ion (and electron) beams. Significant capability has been incorporated in Warp, allowing nearly all sections of an accelerator to be modeled, beginning with the source. Warp has as its core an explicit, three‐dimensional, particle‐in‐cell model. Alongside this is a rich set of tools for describing the applied fields of the accelerator lattice, and embedded conducting surfaces (which are captured at sub‐grid resolution). Also incorporated are models with reduced dimensionality: an axisymmetric model and a transverse “slice” model. The code takes advantage of modern programming techniques, including object orientation, parallelism, and scripting (via Python). It is at the forefront in the use of the computational technique of adaptive mesh refinement, which has been particularly successful in the area of diode and injector modeling, both steady‐state and time‐dependent. In the presentation, some of the major aspects of Warp will be overviewed, especially those that could be useful in modeling ECR sources. Warp has been benchmarked against both theory and experiment. Recent results will be presented showing good agreement of Warp with experimental results from the STS500 injector test stand. Additional information can be found on the web page http://hif.lbl.gov/theory/WARP{\_}summary.html.}, -author = {Grote, David P. and Friedman, Alex and Vay, Jean Luc and Haber, Irving}, -booktitle = {AIP Conference Proceedings}, -pages = {55--58}, -title = {{The WARP code: Modeling high intensity ion beams}}, -volume = {749}, -year = {2005} -} -@article{Villasenorcpc92, -author = {Villasenor, J and Buneman, O}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -number = {2-3}, -pages = {306--316}, -title = {{Rigorous Charge Conservation For Local Electromagnetic-Field Solvers}}, -volume = {69}, -year = {1992} -} -@misc{Spitkovsky:Icnsp2011, -annote = {Private Communication}, -author = {Sironi, L and Spitkovsky, A}, -title = {{No Title}}, -year = {2011} -} -@article{LeemansPRL2014, -author = {Leemans, W P and Gonsalves, A J and Mao, H.-S. and Nakamura, K and Benedetti, C and Schroeder, C B and T{\'{o}}th, Cs. and Daniels, J and Mittelberger, D E and Bulanov, S S and Vay, J.-L. and Geddes, C G R and Esarey, E}, -doi = {10.1103/PhysRevLett.113.245002}, -journal = {Phys. Rev. Lett.}, -month = {dec}, -number = {24}, -pages = {245002}, -publisher = {American Physical Society}, -title = {{Multi-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping Regime}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.113.245002}, -volume = {113}, -year = {2014} -} -@article{Abejcp86, -author = {Abe, H and Sakairi, N and Itatani, R and Okuda, H}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {apr}, -number = {2}, -pages = {247--267}, -title = {{High-Order Spline Interpolations In The Particle Simulation}}, -volume = {63}, -year = {1986} -} -@article{LeeCPC2015, -author = {Lee, P and Vay, J.-L.}, -doi = {http://dx.doi.org/10.1016/j.cpc.2015.04.004}, -issn = {0010-4655}, -journal = {Computer Physics Communications}, -keywords = {High order FDTD,Perfectly Matched (PML),Pseudo-spectral solvers}, -pages = {1--9}, -title = {{Efficiency of the Perfectly Matched Layer with high-order finite difference and pseudo-spectral Maxwell solvers}}, -url = {http://www.sciencedirect.com/science/article/pii/S0010465515001356}, -volume = {194}, -year = {2015} -} -@article{Vaylpb2002, -author = {Vay, Jl and Colella, P and Mccorquodale, P and {Van Straalen}, B and Friedman, A and Grote, Dp}, -doi = {10.1017/S0263034602204139}, -issn = {0263-0346}, -journal = {Laser And Particle Beams}, -month = {dec}, -number = {4}, -pages = {569--575}, -title = {{Mesh Refinement For Particle-In-Cell Plasma Simulations: Applications To And Benefits For Heavy Ion Fusion}}, -volume = {20}, -year = {2002} -} -@inproceedings{Vaypac11, -address = {New-York, Ny, Usa}, -annote = {Wep154}, -author = {Vay, J.-L. and Furman, M.{\~{}}A. and Venturini, M}, -booktitle = {Proc. Particle Accelerator Conference}, -title = {{Direct Numerical Modeling Of E-Cloud Driven Instability Of A Bunch Train In The Cern Sps}}, -url = {Http://accelconf.web.cern.ch/Accelconf/Pac2011/Papers/Wep154.Pdf}, -year = {2011} -} -@article{Tzoufrasprl2008, -author = {Tzoufras, M and Lu, W and Tsung, F S and Huang, C and Mori, W B and Katsouleas, T and Vieira, J and Fonseca, R A and Silva, L O}, -doi = {10.1103/Physrevlett.101.145002}, -journal = {Physical Review Letters}, -month = {oct}, -number = {14}, -pages = {145002}, -title = {{Beam Loading In The Nonlinear Regime Of Plasma-Based Acceleration Rid C-6436-2009 Rid B-7680-2009 Rid C-3169-2009}}, -volume = {101}, -year = {2008} -} -@article{Bulanovphysfluid1992, -author = {Bulanov, S V and Inovenkov, I N and Kirsanov, V I and Naumova, N M and Sakharov, A S}, -doi = {10.1063/1.860046}, -journal = {Physics Of Fluids B-Plasma Physics}, -month = {jul}, -number = {7}, -pages = {1935--1942}, -title = {{Nonlinear Depletion Of Ultrashort And Relativistically Strong Laser-Pulses In An Underdense Plasma}}, -volume = {4}, -year = {1992} -} -@article{Martinsnaturephysics10, -author = {Martins, S F and Fonseca, R A and Lu, W and Mori, W B and Silva, L O}, -doi = {10.1038/Nphys1538}, -issn = {1745-2473}, -journal = {Nature Physics}, -month = {apr}, -number = {4}, -pages = {311--316}, -title = {{Exploring Laser-Wakefield-Accelerator Regimes For Near-Term Lasers Using Particle-In-Cell Simulation In Lorentz-Boosted Frames}}, -volume = {6}, -year = {2010} -} -@article{Friedmanpop10, -author = {Friedman, A and Barnard, J J and Cohen, R H and Grote, D P and Lund, S M and Sharp, W M and Faltens, A and Henestroza, E and Jung, J.-Y. and Kwan, J W and Lee, E P and Leitner, M A and Logan, B G and Vay, J.-L. and Waldron, W L and Davidson, R C and Dorf, M and Gilson, E P and Kaganovich, I D}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -pages = {056704 (9 Pp.)}, -title = {{Beam Dynamics Of The Neutralized Drift Compression Experiment-Ii, A Novel Pulse-Compressing Ion Accelerator}}, -volume = {17}, -year = {2010} -} -@article{Schroederpop2006, -author = {Schroeder, C B and Esarey, E and Shadwick, B A and Leemans, W P}, -doi = {10.1063/1.2173960}, -journal = {Physics Of Plasmas}, -month = {mar}, -number = {3}, -pages = {33103}, -title = {{Trapping, Dark Current, And Wave Breaking In Nonlinear Plasma Waves}}, -volume = {13}, -year = {2006} -} -@article{Friedmanpfb92, -annote = {33Rd Annual Meeting Of The Division Of Plasma Physics Of The American Physical Soc, Tampa, Fl, Nov 04-08, 1991}, -author = {Friedman, A and Grote, Dp and Haber, I}, -doi = {10.1063/1.860024}, -institution = {Amer Phys Soc, Div Plasma Phys}, -issn = {0899-8221}, -journal = {Physics Of Fluids B-Plasma Physics}, -month = {jul}, -number = {7, Part 2}, -pages = {2203--2210}, -title = {{3-Dimensional Particle Simulation Of Heavy-Ion Fusion Beams}}, -volume = {4}, -year = {1992} -} -@book{Gustafssonkreissoliger, -author = {Gustafsson, B and Kreiss, H.-O. and Oliger, J}, -publisher = {Wiley}, -title = {{Time Dependent Problems And Difference Methods}}, -year = {1995} -} -@article{Winklehnerji2010, -author = {Winklehner, D and Todd, D and Benitez, J and Strohmeier, M and Grote, D and Leitner, D}, -doi = {10.1088/1748-0221/5/12/P12001}, -issn = {1748-0221}, -journal = {Journal of Instrumentation}, -month = {dec}, -title = {{Comparison Of Extraction And Beam Transport Simulations With Emittance Measurements From The Ecr Ion Source Venus}}, -volume = {5}, -year = {2010} -} -@inproceedings{Vaypac09, -address = {Vancouver, Canada}, -annote = {Tu1Pbi04}, -author = {{Vay $\backslash$it Et Al.}, J.-L.}, -booktitle = {Proc. Particle Accelerator Conference}, -title = {{Application Of The Reduction Of Scale Range In A Lorentz Boosted Frame To The Numerical Simulation Of Particle Acceleration Devices}}, -year = {2009} -} -@article{Vincenti2016a, -abstract = {Very high order or pseudo-spectral Maxwell solvers are the method of choice to reduce discretization effects (e.g. numerical dispersion) that are inherent to low order Finite-Difference Time-Domain (FDTD) schemes. However, due to their large stencils, these solvers are often subject to truncation errors in many electromagnetic simulations. These truncation errors come from non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the simulation results. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solvers and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation. (C) 2015 Elsevier B.V. All rights reserved.}, -author = {Vincenti, H. and Vay, J.-L.}, -doi = {10.1016/j.cpc.2015.11.009}, -issn = {00104655}, -journal = {Computer Physics Communications}, -keywords = {3D electromagnetic simulations,ABSORPTION,ALGORITHM,APPROXIMATIONS,CLOSED-FORM EXPRESSIONS,Domain decomposition technique,Effects of stencil truncation errors,PERFECTLY MATCHED LAYER,Perfectly Matched Layers,Pseudo-spectral Maxwell solver,SIMULATIONS,TAYLOR-SERIES,Very high-order Maxwell solver,WAVES}, -month = {mar}, -pages = {147--167}, -publisher = {ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS}, -title = {{Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference Maxwell solver}}, -url = {https://apps.webofknowledge.com/full{\_}record.do?product=UA{\&}search{\_}mode=GeneralSearch{\&}qid=1{\&}SID=1CanLFIHrQ5v8O7cxqV{\&}page=1{\&}doc=2}, -volume = {200}, -year = {2016} -} -@article{Vayjcp02, -author = {Vay, J.-L.}, -doi = {10.1006/Jcph.2002.7175}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {dec}, -number = {2}, -pages = {367--399}, -title = {{Asymmetric Perfectly Matched Layer For The Absorption Of Waves}}, -volume = {183}, -year = {2002} -} -@book{Felsenmarcuvitz, -author = {Felsen, L.{\~{}}B. and Marcuvitz, N}, -publisher = {Ieee Press}, -title = {{Radiation And Scattering Of Waves}}, -year = {1994} -} -@book{Geddesdissertation05, -address = {Berkeley Ca}, -author = {Geddes, C G R}, -publisher = {Ph.D. Dissertation, University Of California, Berkeley}, -title = {{Plasma Channel Guided Laser Wakefield Accelerator}}, -year = {2005} -} -@article{Tsungpop06, -author = {Tsung, Fs and Lu, W and Tzoufras, M and Mori, Wb and Joshi, C and Vieira, Jm and Silva, Lo and Fonseca, Ra}, -doi = {10.1063/1.2198535}, -issn = {1070-664X}, -journal = {Physics Of Plasmas}, -month = {may}, -number = {5}, -pages = {56708}, -title = {{Simulation Of Monoenergetic Electron Generation Via Laser Wakefield Accelerators For 5-25 Tw Lasers}}, -volume = {13}, -year = {2006} -} -@inproceedings{INFERNO, -address = {Rostock-Warnemünde, Germany}, -author = {Benedetti, Carlo and Schroeder, Carl B. and Esarey, Eric and Leemans, Wim P.}, -booktitle = {ICAP}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Benedetti et al. - 2012 - Efficient Modeling of Laser-plasma Accelerators Using the Ponderomotive-based Code INF{\&}ampRNO.pdf:pdf}, -pages = {THAAI2}, -publisher = {Jacow}, -title = {{Efficient Modeling of Laser-plasma Accelerators Using the Ponderomotive-based Code INF{\&}RNO}}, -url = {http://accelconf.web.cern.ch/AccelConf/ICAP2012/papers/thaai2.pdf}, -year = {2012} -} -@article{GonsalvesNP2011, -author = {Gonsalves, A J and Nakamura, K and Lin, C and Panasenko, D and Shiraishi, S and Sokollik, T and Benedetti, C and Schroeder, C B and Geddes, C G R and van Tilborg, J and Osterhoff, J and Esarey, E and Toth, C and Leemans, W P}, -doi = {10.1038/NPHYS2071}, -issn = {1745-2473}, -journal = {NATURE PHYSICS}, -month = {nov}, -number = {11}, -pages = {862--866}, -title = {{Tunable laser plasma accelerator based on longitudinal density tailoring}}, -volume = {7}, -year = {2011} -} -@article{Ohtsuboprstab2010, -author = {Ohtsubo, S and Fujioka, M and Higaki, H and Ito, K and Okamoto, H and Sugimoto, H and Lund, S M}, -doi = {10.1103/Physrevstab.13.044201}, -issn = {1098-4402}, -journal = {Physical Review Special Topics-Accelerators And Beams}, -month = {apr}, -number = {4}, -title = {{Experimental Study Of Coherent Betatron Resonances With A Paul Trap}}, -volume = {13}, -year = {2010} -} -@inproceedings{Warp, -author = {Grote, D P and Friedman, A and Vay, J.-L. and Haber, I}, -booktitle = {Aip Conference Proceedings}, -issn = {0094-243X}, -number = {749}, -pages = {55--58}, -title = {{The Warp Code: Modeling High Intensity Ion Beams}}, -year = {2005} -} -@article{Mccorquodalejcp2004, -author = {Mccorquodale, P and Colella, P and Grote, Dp and Vay, Jl}, -doi = {10.1016/J.Jcp.2004.04.022}, -issn = {0021-9991}, -journal = {Journal of Computational Physics}, -month = {nov}, -number = {1}, -pages = {34--60}, -title = {{A Node-Centered Local Refinement Algorithm For Poisson's Equation In Complex Geometries}}, -volume = {201}, -year = {2004} -} -@article{Londrillo2010, -abstract = {In this paper we present some new results on our investigation aimed at extending to higher order (HOPIC) the classical PIC framework. After reviewing the basic resolution properties of the Runge–Kutta time integrator, coupled to fourth (sixth) order compact schemes for space derivatives in the Maxwell equations, we focus on the problem of extending charge conservation schemes to a general HOPIC framework. This issue represents the main contribution of the present work. We consider then a few numerical examples of 1D laser-plasma interaction in the under-dense and over-dense regimes relevant for ions acceleration, to test grid convergence and to compare HOPIC results with standard PIC schemes (LOPIC).}, -author = {Londrillo, P. and Benedetti, C. and Sgattoni, A.}, -doi = {10.1016/j.nima.2010.01.055}, -file = {:Users/jlvay/Library/Application Support/Mendeley Desktop/Downloaded/Londrillo, Benedetti, Sgattoni - 2010 - Charge preserving high order PIC schemes.pdf:pdf}, -issn = {01689002}, -journal = {Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment}, -number = {1}, -pages = {28--35}, -title = {{Charge preserving high order PIC schemes}}, -volume = {620}, -year = {2010} -} -@article{GodfreyJCP2014_FDTD, -abstract = {A procedure for largely suppressing the numerical Cherenkov instability in finite difference time-domain (FDTD) particle-in-cell (PIC) simulations of cold, relativistic beams is derived, and residual growth rates computed and compared with WARP code simulation results. Sample laser-plasma acceleration simulation output is provided to further validate the new procedure.}, -author = {Godfrey, Brendan B. and Vay, Jean Luc}, -journal = {Journal of Computational Physics}, -keywords = {Finite difference time-domain,Numerical stability,Particle-in-cell,Relativistic beam}, -pages = {1--6}, -title = {{Suppressing the numerical Cherenkov instability in FDTD PIC codes}}, -volume = {267}, -year = {2014} -} -@article{Shortley-Weller, -author = {Shortley, Gh and Weller, R}, -doi = {10.1063/1.1710426}, -issn = {0021-8979}, -journal = {Journal of Applied Physics}, -month = {may}, -number = {5}, -pages = {334--348}, -title = {{The Numerical Solution Of Laplace's Equation}}, -volume = {9}, -year = {1938} -} -@article{VayPOPL2011, -author = {Vay, Jl and Geddes, C G R and Cormier-Michel, E and Grote, D P}, -doi = {10.1063/1.3559483}, -journal = {Physics Of Plasmas}, -month = {mar}, -number = {3}, -pages = {30701}, -title = {{Effects Of Hyperbolic Rotation In Minkowski Space On The Modeling Of Plasma Accelerators In A Lorentz Boosted Frame}}, -volume = {18}, -year = {2011} -} - -@article{KirchenARXIV2016, -author = {Kirchen, M. and Lehe, R. and Godfrey, B.~B. and Dornmair, I. and Jalas, S. and Peters, K. and Vay, J.-L. and Maier, A.~R.}, -journal = {arXiv:1608.00215}, -title = {{Stable discrete representation of relativistically drifting plasmas}}, -year = {2016} -} - -@article{LeheARXIV2016, -author = {Lehe, R. and Kirchen, M. and Godfrey, B.~B. and Maier, A.~R. and Vay, J.-L.}, -journal = {arXiv:1608.00227}, -title = {{Elimination of Numerical Cherenkov Instability in flowing-plasma Particle-In-Cell simulations by using Galilean coordinates}}, -year = {2016} -} - -@book{godfrey1985iprop, - title={The IPROP Three-Dimensional Beam Propagation Code}, - author={Godfrey, B.B.}, - url={https://books.google.com/books?id=hos\_OAAACAAJ}, - year={1985}, - publisher={Defense Technical Information Center} -} - diff --git a/Docs/source/theory/newcommands.tex b/Docs/source/theory/newcommands.tex deleted file mode 100644 index fe1fe8c84..000000000 --- a/Docs/source/theory/newcommands.tex +++ /dev/null @@ -1,73 +0,0 @@ - -\usepackage{bm} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{graphicx} -\usepackage{url} -\usepackage{hyperref} - -\usepackage[displaymath]{lineno}\usepackage{bm}% bold math - -\newcommand{\fe}{\mathbf{\tilde{E}}} -\newcommand{\fb}{\mathbf{\tilde{B}}} -\newcommand{\fj}{\mathbf{\tilde{J}}} -\newcommand{\ff}{\tilde{F}} -\newcommand{\fg}{\tilde{G}} -\newcommand{\fk}{\mathbf{k}} -\newcommand{\fkhat}{\mathbf{\hat{k}}} - -% Definitions from Remi's paper on Galilean math -\newcommand{\Km}{\vec{K}_{\vec{m}}} -\newcommand{\km}{\vec{k}_{\vec{m}}} -\renewcommand{\vec}[1]{\boldsymbol{#1}} -\newcommand{\vgal}{\vec{v}_{gal}} -\newcommand{\nab}{\vec{\nabla'}} -\newcommand{\Dt}[1]{ \frac{\partial #1}{\partial t}} -\newcommand{\mc}[1]{\hat{\mathcal{#1}}} -\newcommand{\xj}{\vec{x}'_{\vec{j}}} -\newcommand{\Xll}{\vec{X}_{\vec{\ell}}} -\newcommand{\Integ}[1]{\int_{-\infty}^{\infty} \!\!\!\!\!\! - \mathrm{d}#1} -\newcommand{\RInteg}[1]{\int_{0}^{\infty} \!\! \frac{#1\mathrm{d}#1}{(2\pi)^2}} - -% Definitions from Remi's Thesis -\newcommand{\h}{\mathcal{H}} -\newcommand{\hf}{\frac{1}{2}} -\newcommand{\um}{$\mu$m} -\newcommand{\Um}{\mu \mathrm{m}} -\newcommand{\aal}{\langle \vec{a}_l^2 \rangle} -\newcommand{\etad}{ \eta_d } -\newcommand{\etae}{ \eta_\epsilon } -\newcommand{\etag}{ \eta_\gamma } -\newcommand{\tlambda}{ \tilde{\lambda} } -%\newcommand\comment[1]{\textcolor{red}{\textbf{#1}}} -\newcommand{\gsim}{\mathrel{\hbox{\rlap{\lower.55ex -\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$>$}}}} -\newcommand{\lsim}{\mathrel{\hbox{\rlap{\lower.55ex -\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}} -\newcommand{\kfoc}{k_\mathrm{foc}} -\newcommand{\bkfoc}{\bar{k}_\mathrm{foc}} -\newcommand{\xil}{\xi_{\mathrm{laser}}} - -\newcommand{\Ex}[2]{{E_x}^{#1}_{#2}} -\newcommand{\Ey}[2]{{E_y}^{#1}_{#2}} -\newcommand{\Ez}[2]{{E_z}^{#1}_{#2}} -\newcommand{\Bx}[2]{{B_x}^{#1}_{#2}} -\newcommand{\By}[2]{{B_y}^{#1}_{#2}} -\newcommand{\Bz}[2]{{B_z}^{#1}_{#2}} -\newcommand{\Jx}[2]{{J_x}^{#1}_{#2}} -\newcommand{\Jy}[2]{{J_y}^{#1}_{#2}} -\newcommand{\Jz}[2]{{J_z}^{#1}_{#2}} - -\newcommand{\tEr}[2]{\tilde{E_r}^{#1}_{#2}} -\newcommand{\tEt}[2]{\tilde{E_\theta}^{#1}_{#2}} -\newcommand{\tEz}[2]{\tilde{E_z}^{#1}_{#2}} -\newcommand{\tBr}[2]{\tilde{B_r}^{#1}_{#2}} -\newcommand{\tBt}[2]{\tilde{B_\theta}^{#1}_{#2}} -\newcommand{\tBz}[2]{\tilde{B_z}^{#1}_{#2}} -\newcommand{\tJr}[2]{\tilde{J_r}^{#1}_{#2}} -\newcommand{\tJt}[2]{\tilde{J_\theta}^{#1}_{#2}} -\newcommand{\tJz}[2]{\tilde{J_z}^{#1}_{#2}} - -\newcommand{\CCirc}{\textsc{Calder Circ}} -\newcommand{\CCart}{\textsc{Calder 3D}}
\ No newline at end of file diff --git a/Docs/source/theory/theory.rst b/Docs/source/theory/theory.rst new file mode 100644 index 000000000..6493aa546 --- /dev/null +++ b/Docs/source/theory/theory.rst @@ -0,0 +1,13 @@ +Theoretical background +====================== + +This page contains information on the algorithms that are used in WarpX. + +**Topics:** + +.. toctree:: + :maxdepth: 1 + + intro + picsar_theory + warpx_theory diff --git a/Docs/source/theory/theory.tex b/Docs/source/theory/theory.tex deleted file mode 100644 index 8ffb6cf47..000000000 --- a/Docs/source/theory/theory.tex +++ /dev/null @@ -1,873 +0,0 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% Reviews of Accelerator Science and Technology -%% Trim Size: 11in x 8.5in -%% Text Area: 9.25in (include runningheads) x 6.6in -%% Main Text: 10/13pt -%% Date: 16-04-2014 -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%\documentclass[wsdraft]{ws-rast} % to draw visible frames for the text -%\documentclass[twocolumn]{ws-rast} -\documentclass[]{report} - -\usepackage{bm} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{graphicx} -\usepackage{url} -\usepackage{hyperref} - -\usepackage[displaymath]{lineno}\usepackage{bm}% bold math - -\input{newcommands} - -\begin{document} - -\markboth{J.-L. Vay, R. Lehe}{Simulations for plasma and laser acceleration.} - -\title{WarpX} - -%\author{Jean-Luc Vay, R\'emi Lehe} -%\address{Lawrence Berkeley National Laboratory, Berkeley, California, USA\\ -%\email{jlvay@lbl.gov}} - -%\address{Lawrence Berkeley National Laboratory, Berkeley, California, USA\\ -%\email{rlehe@lbl.gov}} - -\maketitle - -\linenumbers - -\begin{abstract} -Computer simulations have had a profound impact on the design and understanding of past and present plasma acceleration experiments, and will be a key component for turning plasma accelerators from a promising technology into a mainstream scientific tool. In this chapter, we present an overview of the numerical techniques used with the most popular approaches to model plasma-based accelerators: electromagnetic Particle-In-Cell, Quasi-Static, Ponderomotive Guiding Center. -The material that is presented is intended to serve as an introduction to the basics of those approaches, and to advances (some of them very recent) that have pushed the state-of-the-art, such as optimal Lorentz boosted frame, advanced laser envelope solvers and the elimination of numerical Cherenkov instability. The Particle-In-Cell method, which has broader interest and is more standardized, is presented in more depth. Additional topics that are cross-cutting such as azimuthal Fourier decomposition or filtering are also discussed, as well as potential challenges and remedies in the initialization of simulations and output of data. Examples of simulations using the techniques that are presented have been left out of this chapter for conciseness, and because simulation results are best understood when presented together - and contrasted with - theoretical and/or experimental results, as in other chapters of this volume. -\end{abstract} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Introduction} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{figure} -%\begin{centering} -%\includegraphics[trim={6cm 4cm 5cm 5cm},clip,scale=0.5]{Plasma_acceleration_sim.pdf} -%\immediate\write18{curl http://hifweb.lbl.gov/public/WarpX/Figures/test.png > Plasma_acceleration_sim.png} -\includegraphics[scale=0.5]{figures/Plasma_acceleration_sim.png} - -%\par\end{centering} -\caption{\label{fig:Plasma_acceleration_sim} Plasma laser-driven (top) and charged-particles-driven (bottom) acceleration (rendering from 3-D Particle-In-Cell simulations). A laser beam (red and blue disks in top picture) or a charged particle beam (red dots in bottom picture) propagating (from left to right) through an under-dense plasma (not represented) displaces electrons, creating a plasma wakefield that supports very high electric fields (pale blue and yellow). These electric fields, which can be orders of magnitude larger than with conventional techniques, can be used to accelerate a short charged particle beam (white) to high-energy over a very short distance.} -\end{figure} - -Computer simulations have had a profound impact on the design and understanding of past and present plasma acceleration experiments \cite{Tsungpop06,Geddesjp08,Geddesscidac09,Huangscidac09}, with -accurate modeling of wake formation, electron self-trapping and acceleration requiring fully kinetic methods (usually Particle-In-Cell) using large computational resources due to the wide range of space and time scales involved. Numerical modeling complements and guides the design and analysis of advanced accelerators, and can reduce development costs significantly. Despite the major recent experimental successes\cite{LeemansPRL2014,Blumenfeld2007,BulanovSV2014,Steinke2016}, the various advanced acceleration concepts need significant progress to fulfill their potential. To this end, large-scale simulations will continue to be a key component toward reaching a detailed understanding of the complex interrelated physics phenomena at play. - -For such simulations, -the most popular algorithm is the Particle-In-Cell (or PIC) technique, -which represents electromagnetic fields on a grid and particles by -a sample of macroparticles. -However, these simulations are extremely computationally intensive, due to the need to resolve the evolution of a driver (laser or particle beam) and an accelerated beam into a structure that is orders of magnitude longer and wider than the accelerated beam. -Various techniques or reduced models have been developed to allow multidimensional simulations at manageable computational costs: quasistatic approximation \cite{Sprangleprl90,Antonsenprl1992,Krallpre1993,Morapop1997,Quickpic}, -ponderomotive guiding center (PGC) models \cite{Antonsenprl1992,Krallpre1993,Quickpic,Benedettiaac2010,Cowanjcp11}, simulation in an optimal Lorentz boosted frame \cite{Vayprl07,Bruhwileraac08,Vayscidac09,Vaypac09,Martinspac09,VayAAC2010,Martinsnaturephysics10,Martinspop10, Martinscpc10, Vayjcp2011,VayPOPL2011,Vaypop2011,Yu2016}, -expanding the fields into a truncated series of azimuthal modes -\cite{godfrey1985iprop,LifschitzJCP2009,DavidsonJCP2015,Lehe2016,AndriyashPoP2016}, fluid approximation \cite{Krallpre1993,Shadwickpop09,Benedettiaac2010} and scaled parameters \cite{Cormieraac08,Geddespac09}. -% -Many codes have been developed and are used for the modeling of plasma accelerators. -A list of such codes is given in table \ref{table_codes}, with the name of the code, its main characteristics, the web site if existing or a reference, and the availability and license, if known. - -In Section 2 of this chapter, we review the standard methods employed in relativistic electromagnetic Particle-In-Cell (PIC) simulations of plasma accelerators, including the core PIC loop steps (particle push, fields update, current deposition from the particles to the grid and fields gathering from the grid to the particles positions), the use of moving window and Lorentz boosted frame, the numerical Cherenkov instability and its mitigation. The electromagnetic quasistatic approximation is presented in section 3, the ponderomotive guiding center approximation in section 4, and azimuthal Fourier decomposition in section 5. Additional considerations such as filtering and inputs/outputs are discussed respectively in sections 6 and 7. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{The electromagnetic Particle-In-Cell method} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{figure} -%\begin{centering} -\includegraphics[scale=0.6]{figures/PIC.png} -%\par\end{centering} -\caption{\label{fig:PIC} The Particle-In-Cell (PIC) method follows the evolution of a collection of charged macro-particles (positively charged in blue on the left plot, negatively charged in red) that evolve self-consistently with their electromagnetic (or electrostatic) fields. The core PIC algorithm involves four operations at each time step: 1) evolve the velocity and position of the particles using the Newton-Lorentz equations, 2) deposit the charge and/or current densities through interpolation from the particles distributions onto the grid, 3) evolve Maxwell's wave equations (for electromagnetic) or solve Poisson's equation (for electrostatic) on the grid, 4) interpolate the fields from the grid onto the particles for the next particle push. Additional ``add-ons'' operations are inserted between these core operations to account for additional physics (e.g. absorption/emission of particles, addition of external forces to account for accelerator focusing or accelerating component) or numerical effects (e.g. smoothing/filtering of the charge/current densities and/or fields on the grid).} -\end{figure} - -In the electromagnetic Particle-In-Cell method \cite{Birdsalllangdon}, -the electromagnetic fields are solved on a grid, usually using Maxwell's -equations - -\begin{subequations} -\begin{eqnarray} -\frac{\mathbf{\partial B}}{\partial t} & = & -\nabla\times\mathbf{E}\label{Eq:Faraday-1}\\ -\frac{\mathbf{\partial E}}{\partial t} & = & \nabla\times\mathbf{B}-\mathbf{J}\label{Eq:Ampere-1}\\ -\nabla\cdot\mathbf{E} & = & \rho\label{Eq:Gauss-1}\\ -\nabla\cdot\mathbf{B} & = & 0\label{Eq:divb-1} -\end{eqnarray} -\end{subequations} -given here in natural units ($\epsilon_0=\mu_0=c=1$), where $t$ is time, $\mathbf{E}$ and -$\mathbf{B}$ are the electric and magnetic field components, and -$\rho$ and $\mathbf{J}$ are the charge and current densities. The -charged particles are advanced in time using the Newton-Lorentz equations -of motion -\begin{subequations} -\begin{align} -\frac{d\mathbf{x}}{dt}= & \mathbf{v},\label{Eq:Lorentz_x-1}\\ -\frac{d\left(\gamma\mathbf{v}\right)}{dt}= & \frac{q}{m}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right),\label{Eq:Lorentz_v-1} -\end{align} -\end{subequations} -where $m$, $q$, $\mathbf{x}$, $\mathbf{v}$ and $\gamma=1/\sqrt{1-v^{2}}$ - are respectively the mass, charge, position, velocity and relativistic -factor of the particle given in natural units ($c=1$). The charge and current densities are interpolated -on the grid from the particles' positions and velocities, while the -electric and magnetic field components are interpolated from the grid -to the particles' positions for the velocity update. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Particle push} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -A centered finite-difference discretization of the Newton-Lorentz -equations of motion is given by -\begin{subequations} -\begin{align} -\frac{\mathbf{x}^{i+1}-\mathbf{x}^{i}}{\Delta t}= & \mathbf{v}^{i+1/2},\label{Eq:leapfrog_x}\\ -\frac{\gamma^{i+1/2}\mathbf{v}^{i+1/2}-\gamma^{i-1/2}\mathbf{v}^{i-1/2}}{\Delta t}= & \frac{q}{m}\left(\mathbf{E}^{i}+\mathbf{\bar{v}}^{i}\times\mathbf{B}^{i}\right).\label{Eq:leapfrog_v} -\end{align} -\end{subequations} -In order to close the system, $\bar{\mathbf{v}}^{i}$ must be -expressed as a function of the other quantities. The two implementations that have become the most popular are presented below. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Boris relativistic velocity rotation} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Particle_pushers/Boris_pusher.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Vay Lorentz-invariant formulation} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Particle_pushers/Vay_pusher.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Field solve} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Various methods are available for solving Maxwell's equations on a -grid, based on finite-differences, finite-volume, finite-element, -spectral, or other discretization techniques that apply most commonly -on single structured or unstructured meshes and less commonly on multiblock -multiresolution grid structures. In this chapter, we summarize the widespread -second order finite-difference time-domain (FDTD) algorithm, its extension -to non-standard finite-differences as well as the pseudo-spectral -analytical time-domain (PSATD) and pseudo-spectral time-domain (PSTD) -algorithms. Extension to multiresolution (or mesh refinement) PIC -is described in, e.g. \cite{VayCSD12,Vaycpc04}. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Finite-Difference Time-Domain (FDTD)} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Maxwell_solvers/Maxwell_FDTD_solver.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Non-Standard Finite-Difference Time-Domain (NSFDTD)} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Maxwell_solvers/Maxwell_NSFDTD_solver.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{Pseudo Spectral Analytical Time Domain (PSATD)} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Maxwell_solvers/Maxwell_PSATD_solver.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Current deposition} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Deposition/Current_deposition.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Field gather} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{Gather/Field_gather.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Mesh refinement} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{AMR/AMR.tex} - -%%%%%\input{Particle_pushers/Vay_pusher.tex} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Boundary conditions} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\input{PML/PML.tex} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Moving window and optimal Lorentz boosted frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The simulations of plasma accelerators from first principles are extremely computationally intensive, due to the need to resolve the evolution of a driver (laser or particle beam) and an accelerated particle beam into a plasma structure that is orders of magnitude longer and wider than the accelerated beam. As is customary in the modeling of particle beam dynamics in standard particle accelerators, a moving window is commonly used to follow the driver, the wake and the accelerated beam. This results in huge savings, by avoiding the meshing of the entire plasma that is orders of magnitude longer than the other length scales of interest. - -\begin{figure} -%\begin{centering} -\includegraphics[scale=0.6]{figures/Boosted_frame.png} -%\par\end{centering} -\caption{\label{fig:PIC} A first principle simulation of a short driver beam (laser or charged particles) propagating through a plasma that is orders of magnitude longer necessitates a very large number of time steps. Recasting the simulation in a frame of reference that is moving close to the speed of light in the direction of the driver beam leads to simulating a driver beam that appears longer propagating through a plasma that appears shorter than in the laboratory. Thus, this relativistic transformation of space and time reduces the disparity of scales, and thereby the number of time steps to complete the simulation, by orders of magnitude.} -\end{figure} - -Even using a moving window, however, a full PIC simulation of a plasma accelerator can be extraordinarily demanding computationally, as many time steps are needed to resolve the crossing of the short driver beam with the plasma column. As it turns out, choosing an optimal frame of reference that travels close to the speed of light in the direction of the laser or particle beam (as opposed to the usual choice of the laboratory frame) enables speedups by orders of magnitude \cite{Vayprl07,Vaypop2011}. This is a result of the properties of Lorentz contraction and dilation of space and time. In the frame of the laboratory, a very short driver (laser or particle) beam propagates through a much longer plasma column, necessitating millions to tens of millions of time steps for parameters in the range of the BELLA or FACET-II experiments. As sketched in Fig. \ref{fig:PIC}, in a frame moving with the driver beam in the plasma at velocity $v=\beta c$ (where $c$ is the speed of light in vacuum), the beam length is now elongated by $\approx(1+\beta)\gamma$ while the plasma contracts by $\gamma$ (where $\gamma=1/\sqrt{1-\beta^2}$ is the relativistic factor associated with the frame velocity). The number of time steps that is needed to simulate a ``longer'' beam through a ``shorter'' plasma is now reduced by up to $\approx(1+\beta) \gamma^2$ (a detailed derivation of the speedup is given below). - -The modeling of a plasma acceleration stage in a boosted frame -involves the fully electromagnetic modeling of a plasma propagating at near the speed of light, for which Numerical Cerenkov -\cite{Borisjcp73,Habericnsp73} is a potential issue, as explained in more details below. -In addition, for a frame of reference moving in the direction of the accelerated beam (or equivalently the wake of the laser), -waves emitted by the plasma in the forward direction expand -while the ones emitted in the backward direction contract, following the properties of the Lorentz transformation. -If one had to resolve both forward and backward propagating -waves emitted from the plasma, there would be no gain in selecting a frame different from the laboratory frame. However, -the physics of interest for a laser wakefield is the laser driving the wake, the wake, and the accelerated beam. -Backscatter is weak in the short-pulse regime, and does not -interact as strongly with the beam as do the forward propagating waves -which stay in phase for a long period. It is thus often assumed that the backward propagating waves -can be neglected in the modeling of plasma accelerator stages. The accuracy of this assumption has been demonstrated by -comparison between explicit codes which include both forward and backward waves and envelope or quasistatic codes which neglect backward waves -\cite{Geddesjp08,Geddespac09,Cowanaac08}. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Theoretical speedup dependency with the frame boost} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The derivation that is given here reproduces the one given in \cite{Vaypop2011}, where the obtainable speedup is derived as an extension of the formula that was derived earlier\cite{Vayprl07}, taking in addition into account the group velocity of the laser as it traverses the plasma. - -Assuming that the simulation box is a fixed number of plasma periods long, which implies the use (which is standard) of a moving window following -the wake and accelerated beam, the speedup is given by the ratio of the time taken by the laser pulse and the plasma to cross each other, divided by the shortest time scale of interest, that is the laser period. To first order, the wake velocity $v_w$ is set by the 1D group velocity of the laser driver, which in the linear (low intensity) limit, is given by \cite{Esareyrmp09}: - -% -\begin{equation} -v_w/c=\beta_w=\left(1-\frac{\omega_p^2}{\omega^2}\right)^{1/2} -\end{equation} -% -where $\omega_p=\sqrt{(n_e e^2)/(\epsilon_0 m_e)}$ is the plasma frequency, $\omega=2\pi c/\lambda$ is the laser frequency, $n_e$ is the plasma density, $\lambda$ is the laser wavelength in vacuum, $\epsilon_0$ is the permittivity of vacuum, $c$ is the speed of light in vacuum, and $e$ and $m_e$ are respectively the charge and mass of the electron. - -In practice, the runs are typically stopped when the last electron beam macro-particle exits the plasma, and a measure of the total time of the simulation is then given by -% -\begin{equation} -T=\frac{L+\eta \lambda_p}{v_w-v_p} -\end{equation} -% -where $\lambda_p\approx 2\pi c/\omega_p$ is the wake wavelength, $L$ is the plasma length, $v_w$ and $v_p=\beta_p c$ are respectively the velocity of the wake and of the plasma relative to the frame of reference, and $\eta$ is an adjustable parameter for taking into account the fraction of the wake which exited the plasma at the end of the simulation. -For a beam injected into the $n^{th}$ bucket, $\eta$ would be set to $n-1/2$. If positrons were considered, they would be injected half a wake period ahead of the location of the electrons injection position for a given period, and one would have $\eta=n-1$. The numerical cost $R_t$ scales as the ratio of the total time to the shortest timescale of interest, which is the inverse of the laser frequency, and is thus given by -% -\begin{equation} -R_t=\frac{T c}{\lambda}=\frac{\left(L+\eta \lambda_p\right)}{\left(\beta_w-\beta_p\right) \lambda} -\end{equation} -% -In the laboratory, $v_p=0$ and the expression simplifies to -% -\begin{equation} -R_{lab}=\frac{T c}{\lambda}=\frac{\left(L+\eta \lambda_p\right)}{\beta_w \lambda} -\end{equation} -% -In a frame moving at $\beta c$, the quantities become -\begin{eqnarray} -\lambda_p^*&=&\lambda_p/\left[\gamma \left(1-\beta_w \beta\right)\right] \\ -L^*&=&L/\gamma \\ -\lambda^*&=& \gamma\left(1+\beta\right) \lambda\\ -\beta_w^*&=&\left(\beta_w-\beta\right)/\left(1-\beta_w\beta\right) \\ -v_p^*&=&-\beta c \\ -T^*&=&\frac{L^*+\eta \lambda_p^*}{v_w^*-v_p^*} \\ -R_t^*&=&\frac{T^* c}{\lambda^*} = \frac{\left(L^*+\eta \lambda_p^*\right)}{\left(\beta_w^*+\beta\right) \lambda^*} -\end{eqnarray} -where $\gamma=1/\sqrt{1-\beta^2}$. - -The expected speedup from performing the simulation in a boosted frame is given by the ratio of $R_{lab}$ and $R_t^*$ -% -\begin{equation} -S=\frac{R_{lab}}{R_t^*}=\frac{\left(1+\beta\right)\left(L+\eta \lambda_p\right)}{\left(1-\beta\beta_w\right)L+\eta \lambda_p} -\label{Eq_scaling1d0} -\end{equation} - -We note that assuming that $\beta_w\approx1$ (which is a valid approximation for most practical cases of interest) and that $\gamma<<\gamma_w$, this expression is consistent with the expression derived earlier \cite{Vayprl07} for the laser-plasma acceleration case, which states that $R_t^*=\alpha R_t/\left(1+\beta\right)$ with $\alpha=\left(1-\beta+l/L\right)/\left(1+l/L\right)$, where $l$ is the laser length which is generally proportional to $\eta \lambda_p$, and $S=R_t/R_T^*$. However, higher values of $\gamma$ are of interest for maximum speedup, as shown below. - -For intense lasers ($a\sim 1$) typically used for acceleration, the energy gain is limited by dephasing \cite{Schroederprl2011}, which occurs over a scale length $L_d \sim \lambda_p^3/2\lambda^2$. -Acceleration is compromised beyond $L_d$ and in practice, the plasma length is proportional to the dephasing length, i.e. $L= \xi L_d$. In most cases, $\gamma_w^2>>1$, which allows the approximations $\beta_w\approx1-\lambda^2/2\lambda_p^2$, and $L=\xi \lambda_p^3/2\lambda^2\approx \xi \gamma_w^2 \lambda_p/2>>\eta \lambda_p$, so that Eq.(\ref{Eq_scaling1d0}) becomes -% -\begin{equation} -S=\left(1+\beta\right)^2\gamma^2\frac{\xi\gamma_w^2}{\xi\gamma_w^2+\left(1+\beta\right)\gamma^2\left(\xi\beta/2+2\eta\right)} -\label{Eq_scaling1d} -\end{equation} -% -For low values of $\gamma$, i.e. when $\gamma<<\gamma_w$, Eq.(\ref{Eq_scaling1d}) reduces to -% -\begin{equation} -S_{\gamma<<\gamma_w}=\left(1+\beta\right)^2\gamma^2 -\label{Eq_scaling1d_simpl2} -\end{equation} -% -Conversely, if $\gamma\rightarrow\infty$, Eq.(\ref{Eq_scaling1d}) becomes -% -\begin{equation} -S_{\gamma\rightarrow\infty}=\frac{4}{1+4\eta/\xi}\gamma_w^2 -\label{Eq_scaling_gamma_inf} -\end{equation} -% -Finally, in the frame of the wake, i.e. when $\gamma=\gamma_w$, assuming that $\beta_w\approx1$, Eq.(\ref{Eq_scaling1d}) gives -% -\begin{equation} -S_{\gamma=\gamma_w}\approx\frac{2}{1+2\eta/\xi}\gamma_w^2 -\label{Eq_scaling_gamma_wake} -\end{equation} -Since $\eta$ and $\xi$ are of order unity, and the practical regimes of most interest satisfy $\gamma_w^2>>1$, the speedup that is obtained by using the frame of the wake will be near the maximum obtainable value given by Eq.(\ref{Eq_scaling_gamma_inf}). - -Note that without the use of a moving window, the relativistic effects that are at play in the time domain would also be at play in the spatial domain \cite{Vayprl07}, and the $\gamma^2$ scaling would transform to $\gamma^4$. Hence, it is important to use a moving window even in simulations in a Lorentz boosted frame. For very high values of the boosted frame, the optimal velocity of the moving window may vanish (i.e. no moving window) or even reverse. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Numerical Stability and alternate formulation in a Galilean frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -The numerical Cherenkov instability (NCI) \cite{Godfreyjcp74} -is the most serious numerical instability affecting multidimensional -PIC simulations of relativistic particle beams and streaming plasmas -\cite{Martinscpc10,VayAAC2010,Vayjcp2011,Spitkovsky:Icnsp2011,GodfreyJCP2013,XuJCP2013}. -It arises from coupling between possibly numerically distorted electromagnetic modes and spurious -beam modes, the latter due to the mismatch between the Lagrangian -treatment of particles and the Eulerian treatment of fields \cite{Godfreyjcp75}. - -In recent papers the electromagnetic dispersion -relations for the numerical Cherenkov instability were derived and solved for both FDTD \cite{GodfreyJCP2013,GodfreyJCP2014_FDTD} -and PSATD \cite{GodfreyJCP2014_PSATD,GodfreyIEEE2014} algorithms. - -Several solutions have been proposed to mitigate the NCI \cite{GodfreyJCP2014,GodfreyIEEE2014,GodfreyJCP2014_PSATD,GodfreyCPC2015,YuCPC2015,YuCPC2015-Circ}. Although -these solutions efficiently reduce the numerical instability, -they typically introduce either strong smoothing of the currents and -fields, or arbitrary numerical corrections, which are -tuned specifically against the NCI and go beyond the -natural discretization of the underlying physical equation. Therefore, -it is sometimes unclear to what extent these added corrections could impact the -physics at stake for a given resolution. - -For instance, NCI-specific corrections include periodically smoothing -the electromagnetic field components \cite{Martinscpc10}, -using a special time step \cite{VayAAC2010,Vayjcp2011} or -applying a wide-band smoothing of the current components \cite{ - VayAAC2010,Vayjcp2011,VayPOPL2011}. Another set of mitigation methods -involve scaling the deposited -currents by a carefully-designed wavenumber-dependent factor -\cite{GodfreyJCP2014_FDTD,GodfreyIEEE2014} or slightly modifying the -ratio of electric and magnetic fields ($E/B$) before gathering their -value onto the macroparticles -\cite{GodfreyJCP2014_PSATD,GodfreyCPC2015}. -Yet another set of NCI-specific corrections -\cite{YuCPC2015,YuCPC2015-Circ} consists -in combining a small timestep $\Delta t$, a sharp low-pass spatial filter, -and a spectral or high-order scheme that is tuned so as to -create a small, artificial ``bump'' in the dispersion relation -\cite{YuCPC2015}. While most mitigation methods have only been applied -to Cartesian geometry, this last -set of methods (\cite{YuCPC2015,YuCPC2015-Circ}) -has the remarkable property that it can be applied -\cite{YuCPC2015-Circ} to both Cartesian geometry and -quasi-cylindrical geometry (i.e. cylindrical geometry with -azimuthal Fourier decomposition \cite{LifschitzJCP2009,DavidsonJCP2015,Lehe2016}). However, -the use of a small timestep proportionally slows down the progress of -the simulation, and the artificial ``bump'' is again an arbitrary correction -that departs from the underlying physics. - -A new scheme was recently proposed, in \cite{KirchenARXIV2016,LeheARXIV2016}, which -completely eliminates the NCI for a plasma drifting at a uniform relativistic velocity --- with no arbitrary correction -- by simply integrating -the PIC equations in \emph{Galilean coordinates} (also known as -\emph{comoving coordinates}). More precisely, in the new -method, the Maxwell equations \emph{in Galilean coordinates} are integrated -analytically, using only natural hypotheses, within the PSATD -framework (Pseudo-Spectral-Analytical-Time-Domain \cite{Habericnsp73,VayJCP2013}). - -The idea of the proposed scheme is to perform a Galilean change of -coordinates, and to carry out the simulation in the new coordinates: -\begin{equation} -\label{eq:change-var} -\vec{x}' = \vec{x} - \vgal t -\end{equation} -where $\vec{x} = x\,\vec{u}_x + y\,\vec{u}_y + z\,\vec{u}_z$ and -$\vec{x}' = x'\,\vec{u}_x + y'\,\vec{u}_y + z'\,\vec{u}_z$ are the -position vectors in the standard and Galilean coordinates -respectively. - -When choosing $\vgal= \vec{v}_0$, where -$\vec{v}_0$ is the speed of the bulk of the relativistic -plasma, the plasma does not move with respect to the grid in the Galilean -coordinates $\vec{x}'$ -- or, equivalently, in the standard -coordinates $\vec{x}$, the grid moves along with the plasma. The heuristic intuition behind this scheme -is that these coordinates should prevent the discrepancy between the Lagrangian and -Eulerian point of view, which gives rise to the NCI \cite{Godfreyjcp75}. - -An important remark is that the Galilean change of -coordinates (\ref{eq:change-var}) is a simple translation. Thus, when used in -the context of Lorentz-boosted simulations, it does -of course preserve the relativistic dilatation of space and time which gives rise to the -characteristic computational speedup of the boosted-frame technique. - -Another important remark is that the Galilean scheme is \emph{not} -equivalent to a moving window (and in fact the Galilean scheme can be -independently \emph{combined} with a moving window). Whereas in a -moving window, gridpoints are added and removed so as to effectively -translate the boundaries, in the Galilean scheme the gridpoints -\emph{themselves} are not only translated but in this case, the physical equations -are modified accordingly. Most importantly, the assumed time evolution of -the current $\vec{J}$ within one timestep is different in a standard PSATD scheme with moving -window and in a Galilean PSATD scheme \cite{LeheARXIV2016}. - -In the Galilean coordinates $\vec{x}'$, the equations of particle -motion and the Maxwell equations take the form -\begin{subequations} -\begin{align} -\frac{d\vec{x}'}{dt} &= \frac{\vec{p}}{\gamma m} - \vgal \label{eq:motion1} \\ -\frac{d\vec{p}}{dt} &= q \left( \vec{E} + -\frac{\vec{p}}{\gamma m} \times \vec{B} \right) \label{eq:motion2}\\ -\left( \Dt{\;} - \vgal\cdot\nab\right)\vec{B} &= -\nab\times\vec{E} \label{eq:maxwell1}\\ -\frac{1}{c^2}\left( \Dt{\;} - \vgal\cdot\nab\right)\vec{E} &= \nab\times\vec{B} - \mu_0\vec{J} \label{eq:maxwell2} -\end{align} -\end{subequations} -where $\nab$ denotes a spatial derivative with respect to the -Galilean coordinates $\vec{x}'$. - -Integrating these equations from $t=n\Delta -t$ to $t=(n+1)\Delta t$ results in the following update equations (see -\cite{LeheARXIV2016} for the details of the derivation): -% -\begin{subequations} -\begin{align} -% -\fb^{n+1} &= \theta^2 C \fb^n - -\frac{\theta^2 S}{ck}i\vec{k}\times \fe^n \nonumber \\ -& + \;\frac{\theta \chi_1}{\epsilon_0c^2k^2}\;i\vec{k} \times - \fj^{n+1/2} \label{eq:disc-maxwell1}\\ -% -\fe^{n+1} &= \theta^2 C \fe^n - +\frac{\theta^2 S}{k} \,c i\vec{k}\times \fb^n \nonumber \\ -& +\frac{i\nu \theta \chi_1 - \theta^2S}{\epsilon_0 ck} \; \fj^{n+1/2}\nonumber \\ -& - \frac{1}{\epsilon_0k^2}\left(\; \chi_2\;\mc{\rho}^{n+1} - - \theta^2\chi_3\;\mc{\rho}^{n} \;\right) i\vec{k} \label{eq:disc-maxwell2} -% -\end{align} -\end{subequations} -% -where we used the short-hand notations $\fe^n \equiv -% -\fe(\vec{k}, n\Delta t)$, $\fb^n \equiv -\fb(\vec{k}, n\Delta t)$ as well as: -\begin{subequations} -\begin{align} -&C = \cos(ck\Delta t) \quad S = \sin(ck\Delta t) \quad k -= |\vec{k}| \label{eq:def-C-S}\\& -\nu = \frac{\vec{k}\cdot\vgal}{ck} \quad \theta = - e^{i\vec{k}\cdot\vgal\Delta t/2} \quad \theta^* = - e^{-i\vec{k}\cdot\vgal\Delta t/2} \label{eq:def-nu-theta}\\& -\chi_1 = \frac{1}{1 -\nu^2} \left( \theta^* - C \theta + i - \nu \theta S \right) \label{eq:def-chi1}\\& -\chi_2 = \frac{\chi_1 - \theta(1-C)}{\theta^*-\theta} \quad -\chi_3 = \frac{\chi_1-\theta^*(1-C)}{\theta^*-\theta} \label{eq:def-chi23} -\end{align} -\end{subequations} -Note that, in the limit $\vgal=\vec{0}$, -(\ref{eq:disc-maxwell1}) and (\ref{eq:disc-maxwell2}) reduce to the standard PSATD -equations \cite{Habericnsp73}, as expected. -As shown in \cite{KirchenARXIV2016,LeheARXIV2016}, -the elimination of the NCI with the new Galilean integration is verified empirically via PIC simulations of uniform drifting plasmas and laser-driven plasma acceleration stages, and confirmed by a theoretical analysis of the instability. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Axi-symmetry and azimuthal Fourier decomposition} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Although full PIC codes are powerful tools, which capture a wide range -of physical phenomena, they also require large computational ressources. -This is partly due to the use of a 3D Cartesian grid, which -leads to a very large number of grid cells. (Typical 3D simulations of -laser-wakefield acceleration require $\sim 10^6$--$ 10^8$ grid -cells.) For this reason, these algorithms need to be highly parallelized, and -high-resolution simulations can only be run on costly large-scale -computer facilities. However, when the driver is -cylindrically-symmetric, it is possible to take advantage of the -symmetry of the problem to reduce the computational cost of the algorithm \cite{godfrey1985iprop,LifschitzJCP2009,DavidsonJCP2015,Lehe2016}. - -\subsection{Azimuthal decomposition} -Let us consider the fields $\vec{E}$, $\vec{B}$, $\vec{J}$ and $\rho$ - in cylindral coordinates $(r,\theta,z)$, expressed as a Fourier series in $\theta$: -% -\begin{equation} -F(r,\theta,z) = \mathrm{Re}\left[ \sum_{\ell=0}^\infty - \tilde{F}_{\ell}(r,z) e^{-i\ell\theta} \right] -\label{eq:chap2:azimuthal} -\end{equation} -% -\begin{equation} -\mathrm{with} \qquad \tilde{F}_{\ell} = C_\ell \int_0^{2\pi} d\theta -\,F(r,\theta,z)e^{i\ell\theta} \qquad -\label{eq:chap2:Fourier-coeffs} -\end{equation} -\begin{equation} -\mathrm{and} \; -\left \{ \begin{array}{l l} -C_{0} = 1/2\pi &\\ -C_\ell = 1/\pi &\mathrm{for}\,\ell > 0 -\end{array} \right. -\end{equation} -% -where $F$ represents any of the quantities $E_r$, -$E_\theta$, $E_z$, $B_r$, $B_\theta$, $B_z$, $J_r$, $J_\theta$, $J_z$ -are $\rho$, and where the -$\tilde{F}_\ell$ are the associated Fourier components ($\ell$ is the -index of the corresponding azimuthal mode). In the general case, this -azimuthal decomposition does not simplify the problem, since an -infinity of modes have to be considered in (\ref{eq:chap2:azimuthal}). However, in the case of a -cylindrically-symmetric laser pulse, only the very first modes have -non-zero components. For instance, the wakefield is represented -exclusively by the mode $\ell = 0$. (This is because the quantities $E_r$, -$E_\theta$, $E_z$, $B_r$, $B_\theta$, $B_z$, $J_r$, $J_\theta$, $J_z$ -and $\rho$ associated with the -wakefield are independent of $\theta$.) On the other hand, the field -of the laser pulse \emph{does} -depend on $\theta$, in cylindrical coordinates. For example, for a -cylindrically-symmetric pulse propagating along $z$ and polarized along $\vec{e}_\alpha = \cos(\alpha)\vec{e}_x + \sin(\alpha)\vec{e}_y$: -% -\begin{align} -\vec{E} &= E_0(r,z)\vec{e}_\alpha \\ -& = E_0(r,z) [\; \cos(\alpha)(\cos(\theta)\vec{e}_r - \sin(\theta)\vec{e}_\theta) \; \nonumber \\ -& + \; \sin(\alpha)(\sin(\theta)\vec{e}_r + \cos(\theta)\vec{e}_\theta) \; ]\\ -& = \mathrm{Re}[ \; E_0(r,z) e^{i\alpha} e^{-i\theta} \; ]\vec{e}_r \; \nonumber \\ -& + \; \mathrm{Re}[ \; -i E_0(r,z) e^{i\alpha} e^{-i\theta} \; ]\vec{e}_\theta. -\end{align} -% -Here the amplitude $E_0$ does not depend on $\theta$ because the pulse was assumed -to be cylindrically symmetric. In this case, the above relation shows -that the fields $E_r$ and $E_\theta$ of the laser are represented -exclusively by the mode $\ell = 1$. A similar calculation shows that -the same holds for $B_r$ and $B_\theta$. On the whole, only the modes -$\ell = 0$ and $\ell = 1$ are a priori necessary to model -laser-wakefield acceleration. -Under those conditions, the infinite sum in -(\ref{eq:chap2:azimuthal}) is truncated at a chosen $\ell_{max}$. In -principle, $\ell_{max} = 1$ is sufficient for laser-wakefield -acceleration. However, $\ell_{max}$ is kept as a free parameter in the algorithm, in order to verify that -higher modes are negligible, as well as to allow for less-symmetric configurations. -Because codes based on this algorithm are able to take into account the modes with $\ell > 0$, they are said to be -``quasi-cylindrical'' (or ``quasi-3D'' by some authors \cite{DavidsonJCP2015}), in contrast to cylindrical codes, which -assume that all fields are independent of $\theta$, and thus only -consider the mode $\ell = 0$. - -\subsection{Discretized Maxwell equations} When the Fourier expressions -of the fields are injected into the Maxwell equations (written in -cylindrical coordinates), the different azimuthal modes -decouple. In this case, the Maxwell-Amp\`ere and Maxwell-Faraday equations --- which are needed to update the fields in the PIC cycle -- can be written separately -for each azimuthal mode $\ell$: -\begin{subequations} -\begin{align} -\frac{\partial \tilde{B}_{r,\ell} }{\partial t} &= -\frac{i\ell}{r}\tilde{E}_{z,\ell} + \frac{\partial - \tilde{E}_{\theta,\ell}}{\partial z} \\[3mm] -\frac{\partial \tilde{B}_{\theta,\ell} }{\partial t} &= - - \frac{\partial \tilde{E}_{r,\ell}}{\partial z} + \frac{\partial - \tilde{E}_{z,\ell}}{\partial r} \\[3mm] -\frac{\partial \tilde{B}_{z,\ell} }{\partial t} &= -- \frac{1}{r} \frac{\partial (r\tilde{E}_{\theta,\ell})}{\partial r} - \frac{i\ell}{r}\tilde{E}_{r,\ell} \\[3mm] -\frac{1}{c^2} \frac{\partial \tilde{E}_{r,\ell} }{\partial t} &= --\frac{i\ell}{r}\tilde{B}_{z,\ell} - \frac{\partial - \tilde{B}_{\theta,\ell}}{\partial z} - \mu_0 \tilde{J}_{r,\ell} \\[3mm] -\frac{1}{c^2}\frac{\partial \tilde{E}_{\theta,\ell} }{\partial t} &= - \frac{\partial \tilde{B}_{r,\ell}}{\partial z} - \frac{\partial - \tilde{B}_{z,\ell}}{\partial r} - \mu_0 \tilde{J}_{\theta,\ell} \\[3mm] -\frac{1}{c^2}\frac{\partial \tilde{E}_{z,\ell} }{\partial t} &= - \frac{1}{r} \frac{\partial (r\tilde{B}_{\theta,\ell})}{\partial r} + - \frac{i\ell}{r}\tilde{B}_{r,\ell} - \mu_0 \tilde{J}_{z,\ell} -\end{align} -\end{subequations} -%\begin{figure} -%\input{./Chap2/Circ_lattice.tex} -%\caption{Representation of the lattice in \CCirc. The -% table shows at which -% position each component of the fields is defined ($j$,$k$ and -% $n$ are integers ; $\Delta r$ and $\Delta z$ are the -% spatial steps of the grid). The above sketch represents one grid -% cell, and the positions of the fields within it.} -%\label{fig:chap2:Circ_lattice} -%\end{figure} -In order to discretize these equations, each azimuthal mode is -represented on a two-dimensional grid, -%(The two dimensions correspond -%to $r$ and $z$.) \Cref{fig:chap2:Circ_lattice} summarizes the -%positions of the different fields within one grid cell, as well as the -%corresponding notations for these fields. Using these notations, -on which the discretized -Maxwell-Amp\`ere and Maxwell-Faraday equations are given by -%\begin{strip} -%\begin{align*} -% -%\frac{ \tBr{n+\hf}{j,\ell,k+\hf}- \tBr{n-\hf}{j,\ell,k+\hf} -%}{\Delta t} =& \frac{i\,\ell}{j\Delta r}\tEz{n}{j,\ell,k+\hf} + (D_z \tilde{E}_{\theta}^n)_{j,\ell,k+\hf} \\ -% -%\frac{ \tBt{n+\hf}{j+\hf,\ell,k+\hf}- \tBt{n-\hf}{j+\hf,\ell,k+\hf} }{\Delta t} =& -(D_z \tilde{E}_r^n)_{j+\hf,\ell,k+\hf} + (D_r \tilde{E}_z^{n})_{j+\hf,\ell,k+\hf} \\ -% -%\frac{ \tBz{n+\hf}{j+\hf,\ell,k}- \tBz{n-\hf}{j+\hf,\ell,k} }{\Delta t} =& -% -\frac{(j+1)\tEt{n}{j+1,\ell,k} - j\tEt{n}{j,\ell,k}}{(j+\hf)\Delta r} - -%\frac{i\,\ell}{(j+\hf) \Delta r}\tEr{n}{j+\hf,\ell,k} \\ -% -%\frac{ \tEr{n+1}{j+\hf,\ell,k}- \tEr{n}{j+\hf,\ell,k}}{c^2 \Delta t} -%=& -\frac{i\,\ell}{(j+\hf)\Delta r}\tBz{n+\hf}{j+\hf,\ell,k} - (D_z \tilde{B}_{\theta}^{n+\hf})_{j+\hf,\ell,k} - \mu_0\tJr{n+\hf}{j+\hf,\ell,k} \\ -% -%\frac{ \tEt{n+1}{j,\ell,k}- \tEt{n}{j,\ell,k}}{c^2 \Delta t} -%=& (D_z \tilde{B}_r^{n+\hf})_{j,\ell,k} - (D_r \tilde{B}_z^{n+\hf})_{j,\ell,k} - \mu_0\tJt{n+\hf}{j,\ell,k} \\ -% -%\frac{ \tEz{n+1}{j,\ell,k+\hf}- \tEz{n}{j,\ell,k+\hf}}{c^2 \Delta t} -%=& \frac{\left(j+\hf\right)\tBt{n+\hf}{j+\hf,\ell,k+\hf} - \left(j-\hf\right)\tBt{n+\hf}{j-\hf,\ell,k+\hf}}{j\Delta r} \\ -%& \qquad \qquad + \frac{i\,\ell}{j\Delta r}\tBr{n+\hf}{j,\ell,k+\hf} - \mu_0\tJz{n+\hf}{j,\ell,k+\hf} -%\end{align*} -%\end{strip} - -\begin{subequations} -\begin{align} -% -D_{t}\tilde{B}_r|_{j,\ell,k+\hf}^{n} \nonumber -=& \frac{i\,\ell}{j\Delta r}\tEz{n}{j,\ell,k+\hf} \\ -& + D_z \tilde{E}_{\theta}|^n_{j,\ell,k+\hf} \\ -% -D_{t}\tilde{B}_\theta|_{j+\hf,\ell,k+\hf}^{n} \nonumber -=& -D_z \tilde{E}_r|^n_{j+\hf,\ell,k+\hf} \\ -& + D_r \tilde{E}_z|^{n}_{j+\hf,\ell,k+\hf} \\ -% -D_{t}\tilde{B}_z|_{j+\hf,\ell,k}^{n} =& \nonumber - -\frac{(j+1)\tEt{n}{j+1,\ell,k} }{(j+\hf)\Delta r} \\ \nonumber - & +\frac{ j\tEt{n}{j,\ell,k}}{(j+\hf)\Delta r} \\ - & - \frac{i\,\ell}{(j+\hf) \Delta r}\tEr{n}{j+\hf,\ell,k} -\end{align} -\end{subequations} -for the magnetic field components, and -\begin{subequations} -\begin{align} -% -\frac{1}{c^2}D_{t}\tilde{E}_r|_{j+\hf,\ell,k}^{n+\hf} \nonumber -=& -\frac{i\,\ell}{(j+\hf)\Delta r}\tBz{n+\hf}{j+\hf,\ell,k} \\ -& - D_z \tilde{B}_{\theta}|^{n+\hf}_{j+\hf,\ell,k} \nonumber\\ -& - \mu_0\tJr{n+\hf}{j+\hf,\ell,k} \\ -% -\frac{1}{c^2}D_{t}\tilde{E}_\theta|_{j,\ell,k}^{n+\hf} \nonumber -=& D_z \tilde{B}_r|^{n+\hf}_{j,\ell,k} - D_r \tilde{B}_z|^{n+\hf}_{j,\ell,k} \\ -& - \mu_0\tJt{n+\hf}{j,\ell,k} \\ -% -\frac{1}{c^2}D_{t}\tilde{E}_z|_{j,\ell,k+\hf}^{n+\hf} \nonumber -=& \frac{\left(j+\hf\right)\tBt{n+\hf}{j+\hf,\ell,k+\hf} }{j\Delta r} \\ -=& -\frac{\left(j-\hf\right)\tBt{n+\hf}{j-\hf,\ell,k+\hf}}{j\Delta r} \nonumber\\ -& + \frac{i\,\ell}{j\Delta r}\tBr{n+\hf}{j,\ell,k+\hf} \nonumber\\ -& - \mu_0\tJz{n+\hf}{j,\ell,k+\hf} -\end{align} -\end{subequations} -for the electric field components. - -The numerical operator $D_r$ and $D_z$ are defined by -\begin{align*} -(D_r F)_{j',\ell,k'} = \frac{F_{j'+\hf,\ell,k'}-F_{j'-\hf,\ell,k'} }{\Delta r} \\ -(D_z F)_{j',\ell,k'} = \frac{F_{j',\ell,k'+\hf}-F_{j',\ell,k'-\hf} }{\Delta z} \\ -\end{align*} -where $j'$ and $k'$ can be integers or half-integers. Notice -that these discretized Maxwell equations are not valid on-axis (i.e. for $j=0$), due to -singularities in some of the terms. Therefore, on the axis, these equations are replaced by specific boundary conditions, which are based on the symmetry properties of the fields (see \cite{LifschitzJCP2009} for details). - -Compared to a 3D Cartesian calculation with $n_x\times n_y \times n_z$ -grid cells, a quasi-cylindrical calculation with two modes ($l=0$ and $l=1$) -will require only $3 \,n_r \times n_z$ grid cells. Assuming $n_x=n_y=n_r=100$ -as a typical transverse resolution, a quasi-cylindrical calculation is typically -over an order of magnitude less computationally demanding than its 3D Cartesian -equivalent. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Filtering} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -It is common practice to apply digital filtering to the charge or -current density in Particle-In-Cell simulations as a complement or -an alternative to using higher order splines \cite{Birdsalllangdon}. -A commonly used filter in PIC simulations is the three points filter -$\phi_{j}^{f}=\alpha\phi_{j}+\left(1-\alpha\right)\left(\phi_{j-1}+\phi_{j+1}\right)/2$ -where $\phi^{f}$ is the filtered quantity. This filter is called -a bilinear filter when $\alpha=0.5$. Assuming $\phi=e^{jkx}$ and -$\phi^{f}=g\left(\alpha,k\right)e^{jkx}$, the filter gain $g$ is -given as a function of the filtering coefficient $\alpha$ and -the wavenumber $k$ by $g\left(\alpha,k\right)=\alpha+\left(1-\alpha\right)\cos\left(k\Delta x\right)\approx1-\left(1-\alpha\right)\frac{\left(k\Delta x\right)^{2}}{2}+O\left(k^{4}\right)$. -The total attenuation $G$ for $n$ successive applications of filters -of coefficients $\alpha_{1}$...$\alpha_{n}$ is given by $G=\prod_{i=1}^{n}g\left(\alpha_{i},k\right)\approx1-\left(n-\sum_{i=1}^{n}\alpha_{i}\right)\frac{\left(k\Delta x\right)^{2}}{2}+O\left(k^{4}\right)$. -A sharper cutoff in $k$ space is provided by using $\alpha_{n}=n-\sum_{i=1}^{n-1}\alpha_{i}$, -so that $G\approx1+O\left(k^{4}\right)$. Such step is called a ``compensation'' -step \cite{Birdsalllangdon}. For the bilinear filter ($\alpha=1/2$), -the compensation factor is $\alpha_{c}=2-1/2=3/2$. For a succession -of $n$ applications of the bilinear factor, it is $\alpha_{c}=n/2+1$. - -It is sometimes necessary to filter on a relatively wide band of wavelength, -necessitating the application of a large number of passes of the bilinear -filter or on the use of filters acting on many points. The former -can become very intensive computationally while the latter is problematic -for parallel computations using domain decomposition, as the footprint -of the filter may eventually surpass the size of subdomains. A workaround -is to use a combination of filters of limited footprint. A solution -based on the combination of three point filters with various strides -was proposed in \cite{Vayjcp2011} and operates as follows. - -The bilinear filter provides complete suppression of the signal at -the grid Nyquist wavelength (twice the grid cell size). Suppression -of the signal at integer multiples of the Nyquist wavelength can be -obtained by using a stride $s$ in the filter $\phi_{j}^{f}=\alpha\phi_{j}+\left(1-\alpha\right)\left(\phi_{j-s}+\phi_{j+s}\right)/2$ -for which the gain is given by $g\left(\alpha,k\right)=\alpha+\left(1-\alpha\right)\cos\left(sk\Delta x\right)\approx1-\left(1-\alpha\right)\frac{\left(sk\Delta x\right)^{2}}{2}+O\left(k^{4}\right)$. -For a given stride, the gain is given by the gain of the bilinear -filter shifted in k space, with the pole $g=0$ shifted from the wavelength -$\lambda=2/\Delta x$ to $\lambda=2s/\Delta x$, with additional poles, -as given by $sk\Delta x=\arccos\left(\frac{\alpha}{\alpha-1}\right)\pmod{2\pi}$. -The resulting filter is pass band between the poles, but since the -poles are spread at different integer values in k space, a wide band -low pass filter can be constructed by combining filters using different -strides. As shown in \cite{Vayjcp2011}, the successive application -of 4-passes + compensation of filters with strides 1, 2 and 4 has -a nearly equivalent fall-off in gain as 80 passes + compensation of -a bilinear filter. Yet, the strided filter solution needs only 15 -passes of a three-point filter, compared to 81 passes for an equivalent -n-pass bilinear filter, yielding a gain of 5.4 in number of operations -in favor of the combination of filters with stride. The width of the -filter with stride 4 extends only on 9 points, compared to 81 points -for a single pass equivalent filter, hence giving a gain of 9 in compactness -for the stride filters combination in comparison to the single-pass -filter with large stencil, resulting in more favorable scaling with the number -of computational cores for parallel calculations. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%\section{Porting onto new architectures, parallelization, vectorization, mesh refinement} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Inputs and outputs} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Initialization of the plasma columns and drivers (laser or particle beam) is performed via the specification of multidimensional functions that describe the initial state with, if needed, a time dependence, or from reconstruction of distributions based on experimental data. Care is needed when initializing quantities in parallel to avoid double counting and ensure smoothness of the distributions at the interface of computational domains. When the sum of the initial distributions of charged particles is not charge neutral, initial fields are computed using generally a static approximation with Poisson solves accompanied by proper relativistic scalings \cite{Vaypop2008, CowanPRSTAB13}. - -Outputs include dumps of particle and field quantities at regular intervals, histories of particle distributions moments, spectra, etc, and plots of the various quantities. In parallel simulations, the diagnostic subroutines need to handle additional complexity from the domain decomposition, as well as large amount of data that may necessitate data reduction in some form before saving to disk. - -Simulations in a Lorentz boosted frame require additional considerations, as described below. - -\subsection{Inputs and outputs in a boosted frame simulation} -\begin{figure} -% \centering - \includegraphics[width=120mm]{figures/Input.png} - \includegraphics[width=120mm]{figures/Output.png} - \caption{(color online) (top) Snapshot of a particle beam showing ``frozen" (grey spheres) and ``active" (colored spheres) macroparticles traversing the injection plane (red rectangle). (bottom) Snapshot of the beam macroparticles (colored spheres) passing through the background of electrons (dark brown streamlines) and the diagnostic stations (red rectangles). The electrons, the injection plane and the diagnostic stations are fixed in the laboratory plane, and are thus counter-propagating to the beam in a boosted frame. } - \label{Fig_inputoutput} -\end{figure} - -The input and output data are often known from, or compared to, experimental data. Thus, calculating in -a frame other than the laboratory entails transformations of the data between the calculation frame and the laboratory -frame. This section describes the procedures that have been implemented in the Particle-In-Cell framework Warp \cite{Warp} to handle the input and output of data between the frame of calculation and the laboratory frame \cite{Vaypop2011}. Simultaneity of events between two frames is valid only for a plane that is perpendicular to the relative motion of the frame. As a result, the input/output processes involve the input of data (particles or fields) through a plane, as well as output through a series of planes, all of which are perpendicular to the direction of the relative velocity between the frame of calculation and the other frame of choice. - -\subsubsection{Input in a boosted frame simulation} -\paragraph{Particles - } -Particles are launched through a plane using a technique that is generic and applies to Lorentz boosted frame simulations in general, including plasma acceleration, and is illustrated using the case of a positively charged particle beam propagating through a background of cold electrons in an assumed continuous transverse focusing system, leading to a well-known growing transverse ``electron cloud'' instability \cite{Vayprl07}. In the laboratory frame, the electron background is initially at rest and a moving window is used to follow the beam progression. Traditionally, the beam macroparticles are initialized all at once in the window, while background electron macroparticles are created continuously in front of the beam on a plane that is perpendicular to the beam velocity. In a frame moving at some fraction of the beam velocity in the laboratory frame, the beam initial conditions at a given time in the calculation frame are generally unknown and one must initialize the beam differently. However, it can be taken advantage of the fact that the beam initial conditions are often known for a given plane in the laboratory, either directly, or via simple calculation or projection from the conditions at a given time in the labortory frame. Given the position and velocity $\{x,y,z,v_x,v_y,v_z\}$ for each beam macroparticle at time $t=0$ for a beam moving at the average velocity $v_b=\beta_b c$ (where $c$ is the speed of light) in the laboratory, and using the standard synchronization ($z=z'=0$ at $t=t'=0$) between the laboratory and the calculation frames, the procedure for transforming the beam quantities for injection in a boosted frame moving at velocity $\beta c$ in the laboratory is as follows (the superscript $'$ relates to quantities known in the boosted frame while the superscript $^*$ relates to quantities that are know at a given longitudinal position $z^*$ but different times of arrival): - -\begin{enumerate} -\item project positions at $z^*=0$ assuming ballistic propagation -\begin{eqnarray} - t^* &=& \left(z-\bar{z}\right)/v_z \label{Eq:t*}\\ - x^* &=& x-v_x t^* \label{Eq:x*}\\ - y^* &=& y-v_y t^* \label{Eq:y*}\\ - z^* &=& 0 \label{Eq:z*} -\end{eqnarray} -the velocity components being left unchanged, -\item apply Lorentz transformation from laboratory frame to boosted frame -\begin{eqnarray} - t'^* &=& -\gamma t^* \label{Eq:tp*}\\ - x'^* &=& x^* \label{Eq:xp*}\\ - y'^* &=& y^* \label{Eq:yp*}\\ - z'^* &=& \gamma\beta c t^* \label{Eq:zp*}\\ - v'^*_x&=&\frac{v_x^*}{\gamma\left(1-\beta \beta_b\right)} \label{Eq:vxp*}\\ - v'^*_y&=&\frac{v_y^*}{\gamma\left(1-\beta \beta_b\right)} \label{Eq:vyp*}\\ - v'^*_z&=&\frac{v_z^*-\beta c}{1-\beta \beta_b} \label{Eq:vzp*} -\end{eqnarray} -where $\gamma=1/\sqrt{1-\beta^2}$. With the knowledge of the time at which each beam macroparticle crosses the plane into consideration, one can inject each beam macroparticle in the simulation at the appropriate location and time. - -\item synchronize macroparticles in boosted frame, obtaining their positions at a fixed $t'=0$ (before any particle is injected) -\begin{eqnarray} - z' &=& z'^*-\bar{v}'^*_z t'^* \label{Eq:zp} -\end{eqnarray} - This additional step is needed for setting the electrostatic or electromagnetic fields at the plane of injection. In a Particle-In-Cell code, the three-dimensional fields are calculated by solving the Maxwell equations (or static approximation like Poisson, Darwin or other \cite{Vaypop2008}) on a grid on which the source term is obtained from the macroparticles distribution. This requires generation of a three-dimensional representation of the beam distribution of macroparticles at a given time before they cross the injection plane at $z'^*$. This is accomplished by expanding the beam distribution longitudinally such that all macroparticles (so far known at different times of arrival at the injection plane) are synchronized to the same time in the boosted frame. To keep the beam shape constant, the particles are ``frozen'' until they cross that plane: the three velocity components and the two position components perpendicular to the boosted frame velocity are kept constant, while the remaining position component is advanced at the average beam velocity. As particles cross the plane of injection, they become regular ``active'' particles with full 6-D dynamics. - -\end{enumerate} - -Figure \ref{Fig_inputoutput} (top) shows a snapshot of a beam that has passed partly through the injection plane. As the frozen beam macroparticles pass through the injection plane (which moves opposite to the beam in the boosted frame), they are converted to ``active" macroparticles. The charge or current density is accumulated from the active and the frozen particles, thus ensuring that the fields at the plane of injection are consistent. - -\paragraph{Laser - } - -Similarly to the particle beam, the laser is injected through a plane perpendicular to the axis of propagation of the laser (by default $z$). -The electric field $E_\perp$ that is to be emitted is given by the formula -\begin{equation} -E_\perp\left(x,y,t\right)=E_0 f\left(x,y,t\right) \sin\left[\omega t+\phi\left(x,y,\omega\right)\right] -\end{equation} -where $E_0$ is the amplitude of the laser electric field, $f\left(x,y,t\right)$ is the laser envelope, $\omega$ is the laser frequency, $\phi\left(x,y,\omega\right)$ is a phase function to account for focusing, defocusing or injection at an angle, and $t$ is time. By default, the laser envelope is a three-dimensional gaussian of the form -\begin{equation} - f\left(x,y,t\right)=e^{-\left(x^2/2 \sigma_x^2+y^2/2 \sigma_y^2+c^2t^2/2 \sigma_z^2\right)} - \end{equation} - where $\sigma_x$, $\sigma_y$ and $\sigma_z$ are the dimensions of the laser pulse; or it can be defined arbitrarily by the user at runtime. -If $\phi\left(x,y,\omega\right)=0$, the laser is injected at a waist and parallel to the axis $z$. - -If, for convenience, the injection plane is moving at constant velocity $\beta_s c$, the formula is modified to take the Doppler effect on frequency and amplitude into account and becomes -\begin{eqnarray} -E_\perp\left(x,y,t\right)&=&\left(1-\beta_s\right)E_0 f\left(x,y,t\right)\nonumber \\ -&\times& \sin\left[\left(1-\beta_s\right)\omega t+\phi\left(x,y,\omega\right)\right]. -\end{eqnarray} - -The injection of a laser of frequency $\omega$ is considered for a simulation using a boosted frame moving at $\beta c$ with respect to the laboratory. Assuming that the laser is injected at a plane that is fixed in the laboratory, and thus moving at $\beta_s=-\beta$ in the boosted frame, the injection in the boosted frame is given by -\begin{eqnarray} -E_\perp\left(x',y',t'\right)&=&\left(1-\beta_s\right)E'_0 f\left(x',y',t'\right)\nonumber \\ -&\times&\sin\left[\left(1-\beta_s\right)\omega' t'+\phi\left(x',y',\omega'\right)\right]\\ -&=&\left(E_0/\gamma\right) f\left(x',y',t'\right) \nonumber\\ -&\times&\sin\left[\omega t'/\gamma+\phi\left(x',y',\omega'\right)\right] -\end{eqnarray} -since $E'_0/E_0=\omega'/\omega=1/\left(1+\beta\right)\gamma$. - -The electric field is then converted into currents that get injected via a 2D array of macro-particles, with one positive and one dual negative macro-particle for each array cell in the plane of injection, whose weights and motion are governed by $E_\perp\left(x',y',t'\right)$. Injecting using this dual array of macroparticles offers the advantage of automatically including the longitudinal component that arises from emitting into a boosted frame, and to automatically verify the discrete Gauss' law thanks to using charge conserving (e.g. Esirkepov) current deposition scheme \cite{Esirkepovcpc01}. - -\subsubsection{Output in a boosted frame simulation} -Some quantities, e.g. charge or dimensions perpendicular to the boost velocity, are Lorentz invariant. -Those quantities are thus readily available from standard diagnostics in the boosted frame calculations. Quantities that do not fall in this category are recorded at a number of regularly spaced ``stations", immobile in the laboratory frame, at a succession of time intervals to record data history, or averaged over time. A visual example is given on Fig. \ref{Fig_inputoutput} (bottom). Since the space-time locations of the diagnostic grids in the laboratory frame generally do not coincide with the space-time positions of the macroparticles and grid nodes used for the calculation in a boosted frame, some interpolation is performed at runtime during the data collection process. As a complement or an alternative, selected particle or field quantities can be dumped at regular intervals and quantities are reconstructed in the laboratory frame during a post-processing phase. The choice of the methods depends on the requirements of the diagnostics and particular implementations. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Outlook} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -The development of plasma-based accelerators depends critically on high-performance, high-fidelity modeling to capture the full complexity of acceleration processes that develop over a large range of space and time scales. The field will continue to be a driver for pushing the state-of-the-art in the detailed modeling of relativistic plasmas. The modeling of tens of multi-GeV stages, as envisioned for plasma-based high-energy physics colliders, will require further advances in algorithmic, coupled to preparing the codes to take full advantage of the upcoming generation of exascale supercomputers. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Acknowledgments} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -This work was supported by US-DOE Contract DE-AC02-05CH11231. - -This document was prepared as an account of work sponsored in part -by the United States Government. While this document is believed to -contain correct information, neither the United States Government -nor any agency thereof, nor The Regents of the University of California, -nor any of their employees, nor the authors makes any warranty, express -or implied, or assumes any legal responsibility for the accuracy, -completeness, or usefulness of any information, apparatus, product, -or process disclosed, or represents that its use would not infringe -privately owned rights. Reference herein to any specific commercial -product, process, or service by its trade name, trademark, manufacturer, -or otherwise, does not necessarily constitute or imply its endorsement, -recommendation, or favoring by the United States Government or any -agency thereof, or The Regents of the University of California. The -views and opinions of authors expressed herein do not necessarily -state or reflect those of the United States Government or any agency -thereof or The Regents of the University of California.\clearpage{} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\bibliographystyle{ws-rast} -\bibliography{rast_2016_vay,rast_2016_vay2} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% Just a reminder that you may have to run bibtex -%% All of it up to \end{document} can be removed -%% if you don't like the warning. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\IfFileExists{\jobname.bbl}{} {\typeout{} \typeout{{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}} -\typeout{{*}{*} Please run \textquotedbl{}bibtex \jobname\textquotedbl{} -to optain} \typeout{{*}{*} the bibliography and then re-run LaTeX} -\typeout{{*}{*} twice to fix the references!} \typeout{{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}{*}} -\typeout{} } - - - -\end{document}
\ No newline at end of file diff --git a/Docs/source/theory/theory_header.rst b/Docs/source/theory/theory_header.rst deleted file mode 100644 index c3d8d9ab9..000000000 --- a/Docs/source/theory/theory_header.rst +++ /dev/null @@ -1,4 +0,0 @@ -====== -Theory -====== - |
