1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
|
#include "FieldSolver/SpectralSolver/SpectralAlgorithms/AvgGalileanAlgorithm.H"
#include "Utils/WarpXConst.H"
#include <cmath>
using namespace amrex;
/* \brief Initialize coefficients for the update equation */
AvgGalileanAlgorithm::AvgGalileanAlgorithm(const SpectralKSpace& spectral_kspace,
const DistributionMapping& dm,
const int norder_x, const int norder_y,
const int norder_z, const bool nodal,
const amrex::Array<amrex::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);
C1_coef = SpectralRealCoefficients(ba, dm, 1, 0);
S1_coef = SpectralRealCoefficients(ba, dm, 1, 0);
C3_coef = SpectralRealCoefficients(ba, dm, 1, 0);
S3_coef = SpectralRealCoefficients(ba, dm, 1, 0);
Psi1_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Psi2_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Psi3_coef = SpectralComplexCoefficients(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);
A1_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
A2_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Rhoold_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Rhonew_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
Jcoef_coef = SpectralComplexCoefficients(ba, dm, 1, 0);
InitializeSpectralCoefficients(spectral_kspace, dm, v_galilean, dt);
}
void AvgGalileanAlgorithm::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<Real> C1 = C1_coef[mfi].array();
Array4<Real> S1 = S1_coef[mfi].array();
Array4<Real> C3 = C3_coef[mfi].array();
Array4<Real> S3 = S3_coef[mfi].array();
Array4<Complex> Psi1 = Psi1_coef[mfi].array();
Array4<Complex> Psi2 = Psi2_coef[mfi].array();
Array4<Complex> Psi3 = Psi3_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();
Array4<Complex> A1 = A1_coef[mfi].array();
Array4<Complex> A2 = A2_coef[mfi].array();
Array4<Complex> CRhoold = Rhoold_coef[mfi].array();
Array4<Complex> CRhonew = Rhonew_coef[mfi].array();
Array4<Complex> Jcoef = Jcoef_coef[mfi].array();
// Extract reals (for portability on GPU)
Real vx = v_galilean[0];
#if (AMREX_SPACEDIM==3)
Real vy = v_galilean[1];
#endif
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 c2 = PhysConst::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);
C1(i,j,k) = std::cos(0.5_rt*c*k_norm*dt);
S1(i,j,k) = std::sin(0.5_rt*c*k_norm*dt);
C3(i,j,k) = std::cos(1.5_rt*c*k_norm*dt);
S3(i,j,k) = std::sin(1.5_rt*c*k_norm*dt);
// 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 = amrex::exp( 0.5_rt*I*kv*dt );
const Complex theta_star = amrex::exp( -0.5_rt*I*kv*dt );
const Complex e_theta = amrex::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._rt/(1._rt-nu*nu) *
(theta_star - C(i,j,k)*theta + I*kv*S_ck(i,j,k)*theta);
Complex C_rho = I* c2 /( (1._rt-theta*theta) * ep0);
Psi1(i,j,k) = theta * ((S1(i,j,k) + I*nu*C1(i,j,k))
- Theta2(i,j,k) * (S3(i,j,k) + I*nu*C3(i,j,k))) /(c*k_norm*dt * (nu*nu - 1._rt));
Psi2(i,j,k) = theta * ((C1(i,j,k) - I*nu*S1(i,j,k))
- Theta2(i,j,k) * (C3(i,j,k) - I*nu*S3(i,j,k))) /(c2*k_norm*k_norm*dt * (nu*nu - 1._rt));
Psi3(i,j,k) = I * theta * (1._rt - theta*theta) /(c*k_norm*dt*nu);
A1(i,j,k) = (Psi1(i,j,k) - 1._rt + I * kv*Psi2(i,j,k) )/ (c2* k_norm*k_norm * (nu*nu - 1._rt));
A2(i,j,k) = (Psi3(i,j,k) - Psi1(i,j,k)) / (c2*k_norm*k_norm);
CRhoold(i,j,k) = C_rho * (theta*theta * A1(i,j,k) - A2(i,j,k));
CRhonew(i,j,k) = C_rho * (A2(i,j,k) - A1(i,j,k));
Jcoef(i,j,k) = (I*kv*A1(i,j,k) + Psi2(i,j,k))/ep0;
// 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._rt - C(i,j,k)))
/(theta_star-theta)/(ep0*k_norm*k_norm);
X3(i,j,k) = (x1 - theta_star*(1._rt - 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._rt - C(i,j,k)) / (ep0*c*c*k_norm*k_norm);
X2(i,j,k) = (1._rt - 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;
Psi1(i,j,k) = (-S1(i,j,k) + S3(i,j,k)) / (c*k_norm*dt);
Psi2(i,j,k) = (-C1(i,j,k) + C3(i,j,k)) / (c2*k_norm*k_norm*dt);
Psi3(i,j,k) = 1._rt;
A1(i,j,k) = (c*k_norm*dt + S1(i,j,k) - S3(i,j,k)) / (c*c2 * k_norm*k_norm*k_norm * dt);
A2(i,j,k) = (c*k_norm*dt + S1(i,j,k) - S3(i,j,k)) / (c*c2 * k_norm*k_norm*k_norm * dt);
CRhoold(i,j,k) = 2._rt * I * S1(i,j,k) * ( dt*C(i,j,k) - S_ck(i,j,k))
/ (c*k_norm*k_norm*k_norm*dt*dt*ep0);
CRhonew(i,j,k) = - I * (c2* k_norm*k_norm * dt*dt - C1(i,j,k) + C3(i,j,k))
/ (c2 * k_norm*k_norm*k_norm*k_norm * ep0 * dt*dt);
Jcoef(i,j,k) = (-C1(i,j,k) + C3(i,j,k)) / (c2*ep0*k_norm*k_norm*dt);
}
if ( nu == 1.) {
X1(i,j,k) = (1._rt - e_theta*e_theta + 2._rt*I*c*k_norm*dt) / (4._rt*c*c*ep0*k_norm*k_norm);
X2(i,j,k) = (3._rt - 4._rt*e_theta + e_theta*e_theta + 2._rt*I*c*k_norm*dt) / (4._rt*ep0*k_norm*k_norm*(1._rt- e_theta));
X3(i,j,k) = (3._rt - 2._rt/e_theta - 2._rt*e_theta + e_theta*e_theta - 2._rt*I*c*k_norm*dt) / (4._rt*ep0*(e_theta - 1._rt)*k_norm*k_norm);
X4(i,j,k) = I*(-1._rt + e_theta*e_theta + 2._rt*I*c*k_norm*dt) / (4._rt*ep0*c*k_norm);
}
} else { // Handle k_norm = 0, by using the analytical limit
C(i,j,k) = 1._rt;
S_ck(i,j,k) = dt;
C1(i,j,k) = 1._rt;
S1(i,j,k) = 0._rt;
C3(i,j,k) = 1._rt;
S3(i,j,k) = 0._rt;
X1(i,j,k) = dt*dt/(2._rt * ep0);
X2(i,j,k) = c2*dt*dt/(6._rt * ep0);
X3(i,j,k) = - c2*dt*dt/(3._rt * ep0);
X4(i,j,k) = -dt/ep0;
Theta2(i,j,k) = 1._rt;
Psi1(i,j,k) = 1._rt;
Psi2(i,j,k) = -dt;
Psi3(i,j,k) = 1._rt;
A1(i,j,k) = 13._rt * dt*dt /24._rt;
A2(i,j,k) = 13._rt * dt*dt /24._rt;
CRhoold(i,j,k) = -I*c2 * dt*dt / (3._rt * ep0);
CRhonew(i,j,k) = -5._rt*I*c2 * dt*dt / (24._rt * ep0);
Jcoef(i,j,k) = -dt/ep0;
}
});
}
};
/* Advance the E and B field in spectral space (stored in `f`)
* over one time step */
void
AvgGalileanAlgorithm::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();
Array4<const Complex> Psi1_arr = Psi1_coef[mfi].array();
Array4<const Complex> Psi2_arr = Psi2_coef[mfi].array();
Array4<const Complex> Psi3_arr = Psi3_coef[mfi].array();
Array4<const Complex> A1_arr = A1_coef[mfi].array();
Array4<const Complex> A2_arr = A2_coef[mfi].array();
Array4<const Complex> Rhonew_arr = Rhonew_coef[mfi].array();
Array4<const Complex> Rhoold_arr = Rhoold_coef[mfi].array();
Array4<const Complex> Jcoef_arr =Jcoef_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 = SpectralAvgFieldIndex;
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);
const Complex Ex_avg = fields(i,j,k,Idx::Ex_avg);
const Complex Ey_avg= fields(i,j,k,Idx::Ey_avg);
const Complex Ez_avg = fields(i,j,k,Idx::Ez_avg);
const Complex Bx_avg = fields(i,j,k,Idx::Bx_avg);
const Complex By_avg = fields(i,j,k,Idx::By_avg);
const Complex Bz_avg = fields(i,j,k,Idx::Bz_avg);
// 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 Real inv_ep0 = 1._rt/PhysConst::ep0;
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);
const Complex Psi1 = Psi1_arr(i,j,k);
const Complex Psi2 = Psi2_arr(i,j,k);
const Complex Psi3 = Psi3_arr(i,j,k);
const Complex A1 = A1_arr(i,j,k);
const Complex A2 = A2_arr(i,j,k);
const Complex CRhoold= Rhoold_arr(i,j,k);
const Complex CRhonew= Rhonew_arr(i,j,k);
const Complex Jcoef = Jcoef_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);
//Update the averaged E,B fields in time on the interval [(n-1/2)dx, (n+1/2)dx]
fields(i,j,k,Idx::Ex_avg) = Psi1*Ex_old
- Psi2*c2*I*(ky*Bz_old - kz*By_old)
+ Jcoef*Jx + ( CRhonew * rho_new + CRhoold*rho_old )*kx;
fields(i,j,k,Idx::Ey_avg) = Psi1*Ey_old
- Psi2*c2*I*(kz*Bx_old - kx*Bz_old)
+ Jcoef*Jy +( CRhonew * rho_new + CRhoold*rho_old )*ky;
fields(i,j,k,Idx::Ez_avg) = Psi1*Ez_old
- Psi2*c2*I*(kx*By_old - ky*Bx_old)
+ Jcoef*Jz + ( CRhonew * rho_new + CRhoold*rho_old )*kz;
fields(i,j,k,Idx::Bx_avg) = Psi1*Bx_old
+ I*Psi2*(ky*Ez_old - kz*Ey_old)
+ A1*I*(ky*Jz - kz*Jy)*inv_ep0;
fields(i,j,k,Idx::By_avg) = Psi1*By_old
+ I*Psi2*(kz*Ex_old - kx*Ez_old)
+ A1*I*(kz*Jx - kx*Jz)*inv_ep0;
fields(i,j,k,Idx::Bz_avg) = Psi1*Bz_old
+ I*Psi2*(kx*Ey_old - ky*Ex_old)
+ A1*I*(kx*Jy - ky*Jx)*inv_ep0;
});
}
};
void
AvgGalileanAlgorithm::CurrentCorrection (SpectralFieldData& field_data,
std::array<std::unique_ptr<amrex::MultiFab>,3>& current,
const std::unique_ptr<amrex::MultiFab>& rho)
{
amrex::Abort("Current correction not implemented for averaged Galilean PSATD");
}
void
AvgGalileanAlgorithm::VayDeposition (SpectralFieldData& field_data,
std::array<std::unique_ptr<amrex::MultiFab>,3>& current)
{
amrex::Abort("Vay deposition not implemented for averaged Galilean PSATD");
}
|
