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#include <GalileanAlgorithm.H>
#include <WarpXConst.H>
#include <cmath>
using namespace amrex;
/* \brief Initialize coefficients for the update equation */
GalileanAlgorithm::GalileanAlgorithm(const SpectralKSpace& spectral_kspace,
const DistributionMapping& dm,
const int norder_x, const int norder_y,
const int norder_z, const bool nodal,
const Array<Real, 3>& v_galilean,
const Real dt)
// Initialize members of base class
: SpectralBaseAlgorithm( spectral_kspace, dm,
norder_x, norder_y, norder_z, nodal )
{
const BoxArray& ba = spectral_kspace.spectralspace_ba;
// Allocate the arrays of coefficients
C_coef = SpectralRealCoefficients(ba, dm, 1, 0);
S_ck_coef = SpectralRealCoefficients(ba, dm, 1, 0);
X1_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
X2_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
X3_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
X4_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Theta2_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
InitializeSpectralCoefficients(spectral_kspace, dm, v_galilean, dt);
};
/* Advance the E and B field in spectral space (stored in `f`)
* over one time step */
void
GalileanAlgorithm::pushSpectralFields(SpectralFieldData& f) const{
// Loop over boxes
for (MFIter mfi(f.fields); mfi.isValid(); ++mfi){
const Box& bx = f.fields[mfi].box();
// Extract arrays for the fields to be updated
Array4<Complex> fields = f.fields[mfi].array();
// Extract arrays for the coefficients
Array4<const Real> C_arr = C_coef[mfi].array();
Array4<const Real> S_ck_arr = S_ck_coef[mfi].array();
Array4<const Complex> X1_arr = X1_coef[mfi].array();
Array4<const Complex> X2_arr = X2_coef[mfi].array();
Array4<const Complex> X3_arr = X3_coef[mfi].array();
Array4<const Complex> X4_arr = X4_coef[mfi].array();
Array4<const Complex> Theta2_arr = Theta2_coef[mfi].array();
// Extract pointers for the k vectors
const Real* modified_kx_arr = modified_kx_vec[mfi].dataPtr();
#if (AMREX_SPACEDIM==3)
const Real* modified_ky_arr = modified_ky_vec[mfi].dataPtr();
#endif
const Real* modified_kz_arr = modified_kz_vec[mfi].dataPtr();
// Loop over indices within one box
ParallelFor(bx,
[=] AMREX_GPU_DEVICE(int i, int j, int k) noexcept
{
// Record old values of the fields to be updated
using Idx = SpectralFieldIndex;
const Complex Ex_old = fields(i,j,k,Idx::Ex);
const Complex Ey_old = fields(i,j,k,Idx::Ey);
const Complex Ez_old = fields(i,j,k,Idx::Ez);
const Complex Bx_old = fields(i,j,k,Idx::Bx);
const Complex By_old = fields(i,j,k,Idx::By);
const Complex Bz_old = fields(i,j,k,Idx::Bz);
// Shortcut for the values of J and rho
const Complex Jx = fields(i,j,k,Idx::Jx);
const Complex Jy = fields(i,j,k,Idx::Jy);
const Complex Jz = fields(i,j,k,Idx::Jz);
const Complex rho_old = fields(i,j,k,Idx::rho_old);
const Complex rho_new = fields(i,j,k,Idx::rho_new);
// k vector values, and coefficients
const Real kx = modified_kx_arr[i];
#if (AMREX_SPACEDIM==3)
const Real ky = modified_ky_arr[j];
const Real kz = modified_kz_arr[k];
#else
constexpr Real ky = 0;
const Real kz = modified_kz_arr[j];
#endif
constexpr Real c2 = PhysConst::c*PhysConst::c;
constexpr Complex I = Complex{0,1};
const Real C = C_arr(i,j,k);
const Real S_ck = S_ck_arr(i,j,k);
const Complex X1 = X1_arr(i,j,k);
const Complex X2 = X2_arr(i,j,k);
const Complex X3 = X3_arr(i,j,k);
const Complex X4 = X4_arr(i,j,k);
const Complex T2 = Theta2_arr(i,j,k);
// Update E (see the original Galilean article)
fields(i,j,k,Idx::Ex) = T2*C*Ex_old
+ T2*S_ck*c2*I*(ky*Bz_old - kz*By_old)
+ X4*Jx - I*(X2*rho_new - T2*X3*rho_old)*kx;
fields(i,j,k,Idx::Ey) = T2*C*Ey_old
+ T2*S_ck*c2*I*(kz*Bx_old - kx*Bz_old)
+ X4*Jy - I*(X2*rho_new - T2*X3*rho_old)*ky;
fields(i,j,k,Idx::Ez) = T2*C*Ez_old
+ T2*S_ck*c2*I*(kx*By_old - ky*Bx_old)
+ X4*Jz - I*(X2*rho_new - T2*X3*rho_old)*kz;
// Update B (see the original Galilean article)
// Note: here X1 is T2*x1/(ep0*c*c*k_norm*k_norm), where
// x1 has the same definition as in the original paper
fields(i,j,k,Idx::Bx) = T2*C*Bx_old
- T2*S_ck*I*(ky*Ez_old - kz*Ey_old)
+ X1*I*(ky*Jz - kz*Jy);
fields(i,j,k,Idx::By) = T2*C*By_old
- T2*S_ck*I*(kz*Ex_old - kx*Ez_old)
+ X1*I*(kz*Jx - kx*Jz);
fields(i,j,k,Idx::Bz) = T2*C*Bz_old
- T2*S_ck*I*(kx*Ey_old - ky*Ex_old)
+ X1*I*(kx*Jy - ky*Jx);
});
}
};
void GalileanAlgorithm::InitializeSpectralCoefficients(const SpectralKSpace& spectral_kspace,
const amrex::DistributionMapping& dm,
const Array<Real, 3>& v_galilean,
const amrex::Real dt)
{
const BoxArray& ba = spectral_kspace.spectralspace_ba;
// Fill them with the right values:
// Loop over boxes and allocate the corresponding coefficients
// for each box owned by the local MPI proc
for (MFIter mfi(ba, dm); mfi.isValid(); ++mfi){
const Box& bx = ba[mfi];
// Extract pointers for the k vectors
const Real* modified_kx = modified_kx_vec[mfi].dataPtr();
#if (AMREX_SPACEDIM==3)
const Real* modified_ky = modified_ky_vec[mfi].dataPtr();
#endif
const Real* modified_kz = modified_kz_vec[mfi].dataPtr();
// Extract arrays for the coefficients
Array4<Real> C = C_coef[mfi].array();
Array4<Real> S_ck = S_ck_coef[mfi].array();
Array4<Complex> X1 = X1_coef[mfi].array();
Array4<Complex> X2 = X2_coef[mfi].array();
Array4<Complex> X3 = X3_coef[mfi].array();
Array4<Complex> X4 = X4_coef[mfi].array();
Array4<Complex> Theta2 = Theta2_coef[mfi].array();
// Extract reals (for portability on GPU)
Real vx = v_galilean[0];
Real vy = v_galilean[1];
Real vz = v_galilean[2];
// Loop over indices within one box
ParallelFor(bx,
[=] AMREX_GPU_DEVICE(int i, int j, int k) noexcept
{
// Calculate norm of vector
const Real k_norm = std::sqrt(
std::pow(modified_kx[i], 2) +
#if (AMREX_SPACEDIM==3)
std::pow(modified_ky[j], 2) +
std::pow(modified_kz[k], 2));
#else
std::pow(modified_kz[j], 2));
#endif
// Calculate coefficients
constexpr Real c = PhysConst::c;
constexpr Real ep0 = PhysConst::ep0;
const Complex I{0.,1.};
if (k_norm != 0){
C(i,j,k) = std::cos(c*k_norm*dt);
S_ck(i,j,k) = std::sin(c*k_norm*dt)/(c*k_norm);
// Calculate dot product with galilean velocity
const Real kv = modified_kx[i]*vx +
#if (AMREX_SPACEDIM==3)
modified_ky[j]*vy +
modified_kz[k]*vz;
#else
modified_kz[j]*vz;
#endif
const Real nu = kv/(k_norm*c);
const Complex theta = MathFunc::exp( 0.5*I*kv*dt );
const Complex theta_star = MathFunc::exp( -0.5*I*kv*dt );
const Complex e_theta = MathFunc::exp( I*c*k_norm*dt );
Theta2(i,j,k) = theta*theta;
if ( (nu != 1.) && (nu != 0) ) {
// Note: the coefficients X1, X2, X3 do not correspond
// exactly to the original Galilean paper, but the
// update equation have been modified accordingly so that
// the expressions/ below (with the update equations)
// are mathematically equivalent to those of the paper.
Complex x1 = 1./(1.-nu*nu) *
(theta_star - C(i,j,k)*theta + I*kv*S_ck(i,j,k)*theta);
// x1, above, is identical to the original paper
X1(i,j,k) = theta*x1/(ep0*c*c*k_norm*k_norm);
// The difference betwen X2 and X3 below, and those
// from the original paper is the factor ep0*k_norm*k_norm
X2(i,j,k) = (x1 - theta*(1 - C(i,j,k)))
/(theta_star-theta)/(ep0*k_norm*k_norm);
X3(i,j,k) = (x1 - theta_star*(1 - C(i,j,k)))
/(theta_star-theta)/(ep0*k_norm*k_norm);
X4(i,j,k) = I*kv*X1(i,j,k) - theta*theta*S_ck(i,j,k)/ep0;
}
if ( nu == 0) {
X1(i,j,k) = (1. - C(i,j,k)) / (ep0*c*c*k_norm*k_norm);
X2(i,j,k) = (1. - S_ck(i,j,k)/dt) / (ep0*k_norm*k_norm);
X3(i,j,k) = (C(i,j,k) - S_ck(i,j,k)/dt) / (ep0*k_norm*k_norm);
X4(i,j,k) = -S_ck(i,j,k)/ep0;
}
if ( nu == 1.) {
X1(i,j,k) = (1. - e_theta*e_theta + 2.*I*c*k_norm*dt) / (4.*c*c*ep0*k_norm*k_norm);
X2(i,j,k) = (3. - 4.*e_theta + e_theta*e_theta + 2.*I*c*k_norm*dt) / (4.*ep0*k_norm*k_norm*(1.- e_theta));
X3(i,j,k) = (3. - 2./e_theta - 2.*e_theta + e_theta*e_theta - 2.*I*c*k_norm*dt) / (4.*ep0*(e_theta - 1.)*k_norm*k_norm);
X4(i,j,k) = I*(-1. + e_theta*e_theta + 2.*I*c*k_norm*dt) / (4.*ep0*c*k_norm);
}
} else { // Handle k_norm = 0, by using the analytical limit
C(i,j,k) = 1.;
S_ck(i,j,k) = dt;
X1(i,j,k) = dt*dt/(2. * ep0);
X2(i,j,k) = c*c*dt*dt/(6. * ep0);
X3(i,j,k) = - c*c*dt*dt/(3. * ep0);
X4(i,j,k) = -dt/ep0;
Theta2(i,j,k) = 1.;
}
});
}
}
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