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3. Ansatz for initial conditions:
phi = ht exp i (omega t - kx x - ky y - kz z)
Ax = hx exp i (omega t - kx x - ky y - kz z)
Ay = hy exp i (omega t - kx x - ky y - kz z)
Az = hz exp i (omega t - kx x - ky y - kz z)
Lorenz gauge:
sinc x = sin x / x
omega^2 = sinc(kx dx/2)^2 kx^2 + sinc(ky dy/2)^2 ky^2 + sinc(kz dz/2)^2 kz^2
discrete div E constraint:
ht omega = -I (ax/dx (-1 + E^(I dx kx)) + ay/dy (-1 + E^(I dy ky)) + az/dz (-1 + E^(I dz kz)))
ht omega = -i (ax/dx (exp i kx dx/2) (2 i sin (kx dx/2)) + ay/dy (exp i ky dy/2) (2 i sin (ky dy/2)) + az/dz (exp i kz dz/2) (2 i sin (kz dz/2)))
ht omega = ax kx (exp i kx dx/2) sinc(kx dx/2) + ay ky (exp i ky dy/2) sinc(ky dy/2)) + az kz (exp i kz dz/2) sinc(kz dz/2)
4. Discrete ansatz:
- Choose E and B field
- Choose phi and A analytically
- Calculate E and B discretely to ensure constraints
Ex = cos (omega t - kz z)
Ey = 0
Ez = 0
Byz = 0
Bzx = cos (omega t - kz z)
Bxy = 0
phi = 0
E = curl C
Cx = 0
Cy = 1/kz sin (omega t - kz z)
Cz = 0
Ax = -1/kz sin (omega t - kz z)
Ay = 0
Az = 0
phi: not present
A: not present
D: faces of primal grid
B: faces of primal grid
E: edges of primal grid (calculated via lossy Hodge dual)
H: edges of primal grid (calculated via lossy Hodge dual)
rho: cells of primal grid
J: faces of primal grid
div D: cells of primal grid
div B: cells of primal grid
# TODO: phi needs no ghost zones
CCTK_REAL phi TYPE=gf TAGS='index={0 0 0} rhs="dtphi"' "Electric potential"
CCTK_REAL dyz TYPE=gf TAGS='index={0 1 1} rhs="dtdyz"' "Electric flux"
CCTK_REAL dzx TYPE=gf TAGS='index={1 0 1} rhs="dtdzx"' "Electric flux"
CCTK_REAL dxy TYPE=gf TAGS='index={1 1 0} rhs="dtdxy"' "Electric flux"
CCTK_REAL ax TYPE=gf TAGS='index={1 0 0} rhs="dtax"' "Magnetic potential"
CCTK_REAL ay TYPE=gf TAGS='index={0 1 0} rhs="dtay"' "Magnetic potential"
CCTK_REAL az TYPE=gf TAGS='index={0 0 1} rhs="dtaz"' "Magnetic potential"
CCTK_REAL byz TYPE=gf TAGS='index={0 1 1} rhs="dtbyz"' "Magnetic flux"
CCTK_REAL bzx TYPE=gf TAGS='index={1 0 1} rhs="dtbzx"' "Magnetic flux"
CCTK_REAL bxy TYPE=gf TAGS='index={1 1 0} rhs="dtbxy"' "Magnetic flux"
CCTK_REAL ex TYPE=gf TAGS='index={1 0 0} rhs="dtex"' "Electric field"
CCTK_REAL ey TYPE=gf TAGS='index={0 1 0} rhs="dtey"' "Electric field"
CCTK_REAL ez TYPE=gf TAGS='index={0 0 1} rhs="dtez"' "Electric field"
CCTK_REAL byz TYPE=gf TAGS='index={0 1 1} rhs="dtbyz"' "Magnetic field"
CCTK_REAL bzx TYPE=gf TAGS='index={1 0 1} rhs="dtbzx"' "Magnetic field"
CCTK_REAL bxy TYPE=gf TAGS='index={1 1 0} rhs="dtbxy"' "Magnetic field"
CCTK_REAL dtphi TYPE=gf TAGS='index={0 0 0} checkpoint="no"' "Electric potential"
CCTK_REAL dtdyz TYPE=gf TAGS='index={0 1 1} checkpoint="no"' "Electric flux RHS"
CCTK_REAL dtdzx TYPE=gf TAGS='index={1 0 1} checkpoint="no"' "Electric flux RHS"
CCTK_REAL dtdxy TYPE=gf TAGS='index={1 1 0} checkpoint="no"' "Electric flux RHS"
CCTK_REAL dtax TYPE=gf TAGS='index={1 0 0} checkpoint="no"' "Magnetic potential"
CCTK_REAL dtay TYPE=gf TAGS='index={0 1 0} checkpoint="no"' "Magnetic potential"
CCTK_REAL dtaz TYPE=gf TAGS='index={0 0 1} checkpoint="no"' "Magnetic potential"
CCTK_REAL dtbyz TYPE=gf TAGS='index={0 1 1} checkpoint="no"' "Magnetic flux RHS"
CCTK_REAL dtbzx TYPE=gf TAGS='index={1 0 1} checkpoint="no"' "Magnetic flux RHS"
CCTK_REAL dtbxy TYPE=gf TAGS='index={1 1 0} checkpoint="no"' "Magnetic flux RHS"
CCTK_REAL curlayz TYPE=gf TAGS='index={0 1 1} checkpoint="no"' "Magnetic potential constraint"
CCTK_REAL curlazx TYPE=gf TAGS='index={1 0 1} checkpoint="no"' "Magnetic potential constraint"
CCTK_REAL curlaxy TYPE=gf TAGS='index={1 1 0} checkpoint="no"' "Magnetic potential constraint"
CCTK_REAL dive TYPE=gf TAGS='index={0 0 0} checkpoint="no"' "Electric constraint"
CCTK_REAL divd TYPE=gf TAGS='index={1 1 1} checkpoint="no"' "Electric constraint"
CCTK_REAL avgphi TYPE=gf TAGS='checkpoint="no"' "Averaged electric potential"
CCTK_REAL avga TYPE=gf TAGS='checkpoint="no"' { avgax avgay avgaz } "Averaged magnetic potential"
CCTK_REAL avge TYPE=gf TAGS='checkpoint="no"' { avgex avgey avgez } "Averaged electric field"
CCTK_REAL avgb TYPE=gf TAGS='checkpoint="no"' { avgbyz avgbzx avgbxy } "Averaged magnetic field"
CCTK_REAL avgcurla TYPE=gf TAGS='checkpoint="no"' { avgcurlyz avgcurlzx avgcurlxy } "Averaged magnetic potential constraint"
CCTK_REAL avgdive TYPE=gf TAGS='checkpoint="no"' "Averaged electric constraint"
CCTK_REAL avgdivb TYPE=gf TAGS='checkpoint="no"' "Averaged magnetic constraint"
# CCTK_REAL phi1 TYPE=gf TAGS='index={0 0 0}' "Electric potential"
# CCTK_REAL dive1 TYPE=gf TAGS='index={0 0 0}' "Electric constraint"
CCTK_REAL avgd TYPE=gf TAGS='checkpoint="no"' { avgdyz avgdzx avgdxy } "Cell-averaged electric flux"
CCTK_REAL avgb TYPE=gf TAGS='checkpoint="no"' { avgbyz avgbzx avgbxy } "Cell-averaged magnetic flux"
} "Set up hydro initial conditions"
# SCHEDULE Maxwell_EstimateError AT postinitial
# {
# LANG: C
# READS: phi(everywhere)
# READS: ax(everywhere) ay(everywhere) az(everywhere)
# READS: ex(everywhere) ey(everywhere) ez(everywhere)
# READS: byz(everywhere) bzx(everywhere) bxy(everywhere)
# WRITES: CarpetX::regrid_error(interior)
# } "Estimate local error for regridding initial conditions"
# SCHEDULE Maxwell_Constraints AT postinitial
# {
# LANG: C
# READS: ax(interior) ay(interior) az(interior)
# READS: ex(everywhere) ey(everywhere) ez(everywhere)
# READS: byz(interior) bzx(interior) bxy(interior)
# WRITES: curlayz(interior) curlazx(interior) curlaxy(interior)
# WRITES: dive(interior)
# WRITES: divb(interior)
# SYNC: curlayz curlazx curlaxy
# SYNC: dive
# SYNC: divb
# } "Calculate constraints"
#
# SCHEDULE Maxwell_Solve AT postinitial AFTER Maxwell_Constraints
# {
# LANG: C
# OPTIONS: global
# READS: dive(interior)
# WRITES: phi1(interior)
# # WRITES: dive1(interior)
# } "Solve div E constraint"
#
# SCHEDULE Maxwell_UpdatePhi AT postinitial AFTER Maxwell_Solve
# {
# LANG: C
# READS: phi(interior)
# READS: phi1(interior)
# WRITES: phi(interior)
# INVALIDATES: phi1(interior)
# # INVALIDATES: dive1(interior)
# INVALIDATES: curlayz(everywhere) curlazx(everywhere) curlaxy(everywhere)
# INVALIDATES: dive(everywhere)
# INVALIDATES: divb(everywhere)
# SYNC: phi
# } "Update electric potential"
# SCHEDULE Maxwell_Average AT postinitial AFTER Maxwell_Constraints
# {
# LANG: C
# READS: phi(interior)
# READS: ax(interior) ay(interior) az(interior)
# READS: ex(interior) ey(interior) ez(interior)
# READS: byz(interior) bzx(interior) bxy(interior)
# READS: curlayz(interior) curlazx(interior) curlaxy(interior)
# READS: dive(interior)
# READS: divb(interior)
# WRITES: avgphi(interior)
# WRITES: avga(interior)
# WRITES: avge(interior)
# WRITES: avgb(interior)
# WRITES: avgcurla(interior)
# WRITES: avgdive(interior)
# WRITES: avgdivb(interior)
# SYNC: avgphi
# SYNC: avga
# SYNC: avge
# SYNC: avgb
# SYNC: avgcurla
# SYNC: avgdive
# SYNC: avgdivb
# } "Average to cell-centred values"
SCHEDULE Maxwell_EstimateError AT postinitial AFTER (Maxwell_Average, Maxwell_Solve)
{
LANG: C
# READS: avgphi(everywhere)
# READS: avga(everywhere)
# READS: avge(everywhere)
# READS: avgb(everywhere)
WRITES: CarpetX::regrid_error(interior)
} "Estimate local error for regridding during evolution"
} "Set up initial conditions"
READS: ax(interior) ay(interior) az(interior)
READS: ex(everywhere) ey(everywhere) ez(everywhere)
READS: byz(interior) bzx(interior) bxy(interior)
WRITES: curlayz(interior) curlazx(interior) curlaxy(interior)
WRITES: dive(interior)
READS: dyz(everywhere) dzx(everywhere) dxy(everywhere)
READS: byz(everywhere) bzx(everywhere) bxy(everywhere)
WRITES: divd(interior)
# SCHEDULE Maxwell_EstimateError AT poststep
# {
# LANG: C
# READS: phi(everywhere)
# READS: ax(everywhere) ay(everywhere) az(everywhere)
# READS: ex(everywhere) ey(everywhere) ez(everywhere)
# READS: byz(everywhere) bzx(everywhere) bxy(everywhere)
# WRITES: CarpetX::regrid_error(interior)
# } "Estimate local error for regridding during evolution"
READS: phi(interior)
READS: ax(interior) ay(interior) az(interior)
READS: ex(interior) ey(interior) ez(interior)
READS: byz(interior) bzx(interior) bxy(interior)
READS: curlayz(interior) curlazx(interior) curlaxy(interior)
READS: dive(interior)
READS: divb(interior)
WRITES: avgphi(interior)
WRITES: avga(interior)
WRITES: avge(interior)
WRITES: avgb(interior)
WRITES: avgcurla(interior)
WRITES: avgdive(interior)
WRITES: avgdivb(interior)
SYNC: avgphi
SYNC: avga
SYNC: avge
READS: dyz(everywhere) dzx(everywhere) dxy(everywhere)
READS: byz(everywhere) bzx(everywhere) bxy(everywhere)
WRITES: avgdyz(interior) avgdzx(interior) avgdxy(interior)
WRITES: avgbyz(interior) avgbzx(interior) avgbxy(interior)
SYNC: avgd
SYNC: avgcurla
SYNC: avgdive
SYNC: avgdivb
} "Average to cell-centred values"
SCHEDULE Maxwell_EstimateError AT poststep AFTER Maxwell_Average
{
LANG: C
# READS: avgphi(everywhere)
# READS: avga(everywhere)
# READS: avge(everywhere)
# READS: avgb(everywhere)
WRITES: CarpetX::regrid_error(interior)
} "Estimate local error for regridding during evolution"
} "Average fluxes"
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#include <loop.hxx>
#include <cctk.h>
#include <cctk_Arguments_Checked.h>
#include <cctk_Parameters.h>
#include <cmath>
namespace Maxwell {
namespace {
template <typename T> T pow2(T x) { return x * x; }
template <typename T>
T lap(const Loop::GF3D<const T, 1, 1, 1> &var, const Loop::PointDesc &p) {
const auto DI = Loop::vect<int, Loop::dim>::unit(0);
const auto DJ = Loop::vect<int, Loop::dim>::unit(1);
const auto DK = Loop::vect<int, Loop::dim>::unit(2);
return fabs(var(p.I - DI) - 2 * var(p.I) + var(p.I + DI)) +
fabs(var(p.I - DJ) - 2 * var(p.I) + var(p.I + DJ)) +
fabs(var(p.I - DK) - 2 * var(p.I) + var(p.I + DK));
}
} // namespace
extern "C" void Maxwell_EstimateError(CCTK_ARGUMENTS) {
DECLARE_CCTK_ARGUMENTS_Maxwell_EstimateError;
DECLARE_CCTK_PARAMETERS;
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgphi_(cctkGH, avgphi);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgax_(cctkGH, avgax);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgay_(cctkGH, avgay);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgaz_(cctkGH, avgaz);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgex_(cctkGH, avgex);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgey_(cctkGH, avgey);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgez_(cctkGH, avgez);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgbyz_(cctkGH, avgbyz);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgbzx_(cctkGH, avgbzx);
// const Loop::GF3D<const CCTK_REAL, 1, 1, 1> avgbxy_(cctkGH, avgbxy);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> regrid_error_(cctkGH, regrid_error);
if (false) {
// Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
// regrid_error_(p.I) = lap(avgphi_, p) + lap(avgax_, p) + lap(avgay_, p)
// +
// lap(avgaz_, p) + lap(avgex_, p) + lap(avgey_, p) +
// lap(avgez_, p) + lap(avgbyz_, p) + lap(avgbzx_, p)
// + lap(avgbxy_, p);
// });
} else if (true) {
Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
auto r = sqrt(pow2(p.x) + pow2(p.y) + pow2(p.z));
regrid_error_(p.I) = 1 / fmax(r, CCTK_REAL(1.0e-10));
});
} else {
assert(0);
}
}
} // namespace Maxwell
#include <loop.hxx>
#include <cctk.h>
#include <cctk_Arguments_Checked.h>
#include <cctk_Parameters.h>
namespace Maxwell {
extern "C" void Maxwell_EstimateError(CCTK_ARGUMENTS) {
DECLARE_CCTK_ARGUMENTS_Maxwell_EstimateError;
DECLARE_CCTK_PARAMETERS;
const Loop::GF3D<const CCTK_REAL, 0, 0, 0> phi_(cctkGH, phi);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ex_(cctkGH, ex);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ey_(cctkGH, ey);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> ez_(cctkGH, ez);
const Loop::GF3D<const CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const Loop::GF3D<const CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const Loop::GF3D<const CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> regrid_error_(cctkGH, regrid_error);
Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
const auto diffx = [&](const auto &var_, int i, int j, int k) {
CCTK_REAL err = 0;
for (int b = 0; b < 2; ++b)
for (int a = 0; a < 2; ++a)
err += fabs(var_(i + 1, j + a, k + b) - var_(i, j + a, k + b));
return err;
};
const auto diffy = [&](const auto &var_, int i, int j, int k) {
CCTK_REAL err = 0;
for (int b = 0; b < 2; ++b)
for (int a = 0; a < 2; ++a)
err += fabs(var_(i + a, j + 1, k + b) - var_(i + a, j, k + b));
return err;
};
const auto diffz = [&](const auto &var_, int i, int j, int k) {
CCTK_REAL err = 0;
for (int b = 0; b < 2; ++b)
for (int a = 0; a < 2; ++a)
err += fabs(var_(i + a, j + b, k + 1) - var_(i + a, j + b, k));
return err;
};
const auto diff000 = [&](const auto &var_) {
return diffx(var_, p.i, p.j, p.k) + diffy(var_, p.i, p.j, p.k) +
diffz(var_, p.i, p.j, p.k);
};
const auto diff100 = [&](const auto &var_) {
return diffx(var_, p.i - 1, p.j, p.k) + diffx(var_, p.i, p.j, p.k) +
diffy(var_, p.i, p.j, p.k) + diffz(var_, p.i, p.j, p.k);
};
const auto diff010 = [&](const auto &var_) {
return diffx(var_, p.i, p.j, p.k) + diffy(var_, p.i, p.j - 1, p.k) +
diffy(var_, p.i, p.j, p.k) + diffz(var_, p.i, p.j, p.k);
};
const auto diff001 = [&](const auto &var_) {
return diffx(var_, p.i, p.j, p.k) + diffy(var_, p.i, p.j, p.k) +
diffz(var_, p.i, p.j, p.k - 1) + diffz(var_, p.i, p.j, p.k);
};
const auto diff011 = [&](const auto &var_) {
return diffx(var_, p.i, p.j, p.k) + diffy(var_, p.i, p.j - 1, p.k) +
diffy(var_, p.i, p.j, p.k) + diffz(var_, p.i, p.j, p.k - 1) +
diffz(var_, p.i, p.j, p.k);
};
const auto diff101 = [&](const auto &var_) {
return diffx(var_, p.i - 1, p.j, p.k) + diffx(var_, p.i, p.j, p.k) +
diffy(var_, p.i, p.j, p.k) + diffz(var_, p.i, p.j, p.k - 1) +
diffz(var_, p.i, p.j, p.k);
};
const auto diff110 = [&](const auto &var_) {
return diffx(var_, p.i - 1, p.j, p.k) + diffx(var_, p.i, p.j, p.k) +
diffy(var_, p.i, p.j - 1, p.k) + diffy(var_, p.i, p.j, p.k) +
diffz(var_, p.i, p.j, p.k);
};
regrid_error_(p.I) = diff000(phi_) + diff100(ax_) + diff010(ay_) +
diff001(az_) + diff100(ex_) + diff010(ey_) +
diff001(ez_) + diff011(byz_) + diff101(bzx_) +
diff110(bxy_);
});
}
} // namespace Maxwell
#include "dual.hxx"
#include <loop.hxx>
#include <cctk.h>
#include <cctk_Arguments_Checked.h>
#include <cctk_Parameters.h>
#include <cassert>
#include <cmath>
#include <complex>
namespace Maxwell {
using namespace std;
namespace {
int bitsign(bool s) { return s ? -1 : +1; }
template <typename T> T pow2(T x) { return x * x; }
template <typename T> T sinc(T x) { return x == T(0) ? T(1) : sin(x) / x; }
template <typename T> complex<T> cis(T x) { return {cos(x), sin(x)}; }
} // namespace
////////////////////////////////////////////////////////////////////////////////
template <typename T> struct potential { T phi, ax, ay, az; };
// Continuous derivative
template <typename F, typename T>
potential<T> calc_dt(const F &f, T t, T x, T y, T z) {
auto ff = f(dual<T>(t, 1), dual<T>(x), dual<T>(y), dual<T>(z));
return {
ff.phi.eps,
ff.ax.eps,
ff.ay.eps,
ff.az.eps,
};
}
// Discrete derivative
template <typename F, typename T>
potential<T> calc_dxc(const F &f, T t, T x, T y, T z, T dx) {
auto fm = f(t, x - dx / 2, y, z);
auto fp = f(t, x + dx / 2, y, z);
return {
.phi = (fp.phi - fm.phi) / dx,
.ax = (fp.ax - fm.ax) / dx,
.ay = (fp.ay - fm.ay) / dx,
.az = (fp.az - fm.az) / dx,
};
}
template <typename F, typename T>
potential<T> calc_dyc(const F &f, T t, T x, T y, T z, T dy) {
auto fm = f(t, x, y - dy / 2, z);
auto fp = f(t, x, y + dy / 2, z);
return {
.phi = (fp.phi - fm.phi) / dy,
.ax = (fp.ax - fm.ax) / dy,
.ay = (fp.ay - fm.ay) / dy,
.az = (fp.az - fm.az) / dy,
};
}
template <typename F, typename T>
potential<T> calc_dzc(const F &f, T t, T x, T y, T z, T dz) {
auto fm = f(t, x, y, z - dz / 2);
auto fp = f(t, x, y, z + dz / 2);
return {
.phi = (fp.phi - fm.phi) / dz,
.ax = (fp.ax - fm.ax) / dz,
.ay = (fp.ay - fm.ay) / dz,
.az = (fp.az - fm.az) / dz,
};
}
////////////////////////////////////////////////////////////////////////////////
// Plane wave implementation
template <typename T>
potential<complex<T> > plane_wave_impl(T t, T x, T y, T z, T dx, T dy, T dz,
T kx, T ky, T kz, T hx, T hy, T hz) {
DECLARE_CCTK_PARAMETERS;
typedef complex<T> CT;
// choose frequency to ensure div E = 0
T omega = sqrt(pow2(sinc(kx * dx / 2) * kx) + pow2(sinc(ky * dy / 2) * ky) +
pow2(sinc(kz * dz / 2) * kz));
// choose amplitude to ensure Lorenz gauge
CT ht = (CT(hx * kx * sinc(kx * dx / 2)) * cis(kx * dx / 2) +
CT(hy * ky * sinc(ky * dy / 2)) * cis(ky * dy / 2) +
CT(hz * kz * sinc(kz * dz / 2)) * cis(kz * dz / 2)) /
CT(omega);
CT u = cis(omega * t - kx * x - ky * y - kz * z);
return {
.phi = ht * u,
.ax = hx * u,
.ay = hy * u,
.az = hz * u,
};
}
// Plane wave
template <typename T>
potential<T> plane_wave(T t, T x, T y, T z, T dx, T dy, T dz) {
DECLARE_CCTK_PARAMETERS;
// wave number
T kx = M_PI * spatial_frequency_x;
T ky = M_PI * spatial_frequency_y;
T kz = M_PI * spatial_frequency_z;
// amplitude
T hx = amplitude_x;
T hy = amplitude_y;
T hz = amplitude_z;
//
auto p = plane_wave_impl(t, x, y, z, dx, dy, dz, kx, ky, kz, hx, hy, hz);
return {
.phi = real(p.phi),
.ax = real(p.ax),
.ay = real(p.ay),
.az = real(p.az),
};
}
// Plane wave with a triangle profile
template <typename T>
potential<T> triangle_wave(T t, T x, T y, T z, T dx, T dy, T dz) {
DECLARE_CCTK_PARAMETERS;
// wave number
T kx = M_PI * spatial_frequency_x;
T ky = M_PI * spatial_frequency_y;
T kz = M_PI * spatial_frequency_z;
// amplitude
T hx = amplitude_x;
T hy = amplitude_y;
T hz = amplitude_z;
//
potential<T> p{0, 0, 0, 0};
for (int i = 0; i < num_coefficients; ++i) {
const int k = 2 * i + 1;
const T kf = k;
const T hf = bitsign(i & 1) / pow2(kf);
const auto pk = plane_wave_impl(t, x, y, z, dx, dy, dz, kf * kx, kf * ky,
kf * kz, hf * hx, hf * hy, hf * hz);
p.phi += imag(pk.phi);
p.ax += imag(pk.ax);
p.ay += imag(pk.ay);
p.az += imag(pk.az);
}
return p;
}
// Plane wave with Gaussian profile (NOT WORKING)
template <typename T> potential<T> gaussian_wave(T t, T x, T y, T z) {
DECLARE_CCTK_PARAMETERS;
T kx = M_PI * spatial_frequency_x;
T ky = M_PI * spatial_frequency_y;
T kz = M_PI * spatial_frequency_z;
T omega = sqrt(pow2(kx) + pow2(ky) + pow2(kz));
T hx = amplitude_x;
T hy = amplitude_y;
T hz = amplitude_z;
T ht =
omega * (hx * kx + hy * ky + hz * kz) / (pow2(kx) + pow2(ky) + pow2(kz));
T u = exp(-pow2(sin(omega * t - kx * x - ky * y - kz * z) / width) / 2);
return {
.phi = ht * u,
.ax = hx * u,
.ay = hy * u,
.az = hz * u,
};
}
////////////////////////////////////////////////////////////////////////////////
extern "C" void Maxwell_Initial(CCTK_ARGUMENTS) {
DECLARE_CCTK_ARGUMENTS_Maxwell_Initial;
DECLARE_CCTK_PARAMETERS;
const CCTK_REAL t = cctk_time;
// const CCTK_REAL dt = CCTK_DELTA_TIME;
const CCTK_REAL dx = CCTK_DELTA_SPACE(0);
const CCTK_REAL dy = CCTK_DELTA_SPACE(1);
const CCTK_REAL dz = CCTK_DELTA_SPACE(2);
const Loop::GF3D<CCTK_REAL, 0, 0, 0> phi_(cctkGH, phi);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> ex_(cctkGH, ex);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> ey_(cctkGH, ey);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> ez_(cctkGH, ez);
const Loop::GF3D<CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const Loop::GF3D<CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const Loop::GF3D<CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const auto loop_setup{[&](const auto &f4) {
const auto f{[&](const auto &p) { return f4(t, p.x, p.y, p.z); }};
const auto dtf{[&](const auto &p) {
return calc_dt(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z);
}};
const auto dxf{[&](const auto &p) {
return calc_dxc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dx);
}};
const auto dyf{[&](const auto &p) {
return calc_dyc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dy);
}};
const auto dzf{[&](const auto &p) {
return calc_dzc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dz);
}};
Loop::loop_int<0, 0, 0>(
cctkGH, [&](const Loop::PointDesc &p) { phi_(p.I) = f(p).phi; });
Loop::loop_int<1, 0, 0>(
cctkGH, [&](const Loop::PointDesc &p) { ax_(p.I) = f(p).ax; });
Loop::loop_int<0, 1, 0>(
cctkGH, [&](const Loop::PointDesc &p) { ay_(p.I) = f(p).ay; });
Loop::loop_int<0, 0, 1>(
cctkGH, [&](const Loop::PointDesc &p) { az_(p.I) = f(p).az; });
Loop::loop_int<1, 0, 0>(cctkGH, [&](const Loop::PointDesc &p) {
ex_(p.I) = -dxf(p).phi - dtf(p).ax;
});
Loop::loop_int<0, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
ey_(p.I) = -dyf(p).phi - dtf(p).ay;
});
Loop::loop_int<0, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
ez_(p.I) = -dzf(p).phi - dtf(p).az;
});
Loop::loop_int<0, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
byz_(p.I) = dyf(p).az - dzf(p).ay;
});
Loop::loop_int<1, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
bzx_(p.I) = dzf(p).ax - dxf(p).az;
});
Loop::loop_int<1, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
bxy_(p.I) = dxf(p).ay - dyf(p).ax;
});
}};
if (CCTK_EQUALS(setup, "plane wave")) {
loop_setup([&](auto t, auto x, auto y, auto z) {
typedef decltype(t) T;
return plane_wave(t, x, y, z, T(dx), T(dy), T(dz));
});
} else if (CCTK_EQUALS(setup, "triangle wave")) {
loop_setup([&](auto t, auto x, auto y, auto z) {
typedef decltype(t) T;
return triangle_wave(t, x, y, z, T(dx), T(dy), T(dz));
});
// } else if (CCTK_EQUALS(setup, "Gaussian wave")) {
// loop_setup([&](auto t, auto x, auto y, auto z) {
// return gaussian_wave(t, x, y, z);
// });
} else {
assert(0);
}
}
} // namespace Maxwell
#include <loop.hxx>
#include <cctk.h>
#include <cctk_Arguments_Checked.h>
#include <cctk_Parameters.h>
namespace Maxwell {
extern "C" void Maxwell_Solve(CCTK_ARGUMENTS) { SolvePoisson(); }
extern "C" void Maxwell_UpdatePhi(CCTK_ARGUMENTS) {
DECLARE_CCTK_ARGUMENTS_Maxwell_UpdatePhi;
DECLARE_CCTK_PARAMETERS;
const Loop::GF3D<CCTK_REAL, 0, 0, 0> phi_(cctkGH, phi);
const Loop::GF3D<const CCTK_REAL, 0, 0, 0> phi1_(cctkGH, phi1);
Loop::loop_int<0, 0, 0>(
cctkGH, [&](const Loop::PointDesc &p) { phi_(p.I) -= phi1_(p.I); });
}
} // namespace Maxwell
#include "dual.hxx"
#include <functional>
namespace Maxwell {
using namespace std;
void TestDual() {
typedef dual<CCTK_REAL> DREAL;
constexpr equal_to<CCTK_REAL> eq;
constexpr equal_to<DREAL> eqd;
static_assert(eq(DREAL().val, 0));
static_assert(eq(DREAL().eps, 0));
static_assert(eq(DREAL(1).val, 1));
static_assert(eq(DREAL(1).eps, 0));
static_assert(eq(DREAL(1, 2).val, 1));
static_assert(eq(DREAL(1, 2).eps, 2));
static_assert(eqd(DREAL(1, 2), DREAL(1, 2)));
static_assert(!eqd(DREAL(1, 2), DREAL(2, 3)));
static_assert(eqd(+DREAL(1, 2), DREAL(1, 2)));
static_assert(eqd(-DREAL(1, 2), DREAL(-1, -2)));
static_assert(eqd(DREAL(1, 2) + DREAL(3, 4), DREAL(4, 6)));
static_assert(eqd(DREAL(1, 2) - DREAL(3, 4), DREAL(-2, -2)));
static_assert(eqd(2 * DREAL(3, 4), DREAL(6, 8)));
static_assert(eqd(DREAL(3, 4) * 2, DREAL(6, 8)));
static_assert(eqd(DREAL(3, 4) / 2, DREAL(1.5, 2)));
static_assert(eqd(DREAL(2, 3) * DREAL(4, 5), DREAL(8, 22)));
static_assert(eqd(sqrt(DREAL(4, 3)), DREAL(2, 0.75)));
}
} // namespace Maxwell
#include <cctk.h>
#include <cctk_Arguments_Checked.h>
#include <cctk_Parameters.h>
#include <cassert>
#include <cmath>
#include <functional>
#include <ostream>
namespace Maxwell {
using namespace std;
template <typename T> struct dual {
T val, eps;
dual(const dual &) = default;
dual(dual &&) = default;
dual &operator=(const dual &) = default;
dual &operator=(dual &&) = default;
constexpr dual() : val(), eps() {}
constexpr dual(const T &x) : val(x), eps() {}
constexpr dual(const T &x, const T &y) : val(x), eps(y) {}
friend constexpr dual operator+(const dual &x) { return {+x.val, +x.eps}; }
friend constexpr dual operator-(const dual &x) { return {-x.val, -x.eps}; }
friend constexpr dual operator+(const dual &x, const dual &y) {
return {x.val + y.val, x.eps + y.eps};
}
friend constexpr dual operator-(const dual &x, const dual &y) {
return {x.val - y.val, x.eps - y.eps};
}
friend constexpr dual operator+(const dual &x, const T &y) {
return {x.val + y, x.eps};
}
friend constexpr dual operator-(const dual &x, const T &y) {
return {x.val - y, x.eps};
}
friend constexpr dual operator*(const dual &x, const T &y) {
return {x.val * y, x.eps * y};
}
friend constexpr dual operator*(const T &x, const dual &y) {
return {x * y.val, x * y.eps};
}
friend constexpr dual operator/(const dual &x, const T &y) {
return {x.val / y, x.eps / y};
}
friend constexpr dual operator/(const dual &x, const dual &y) {
assert(y.eps == 0);
return x / y.val;
}
friend constexpr dual operator*(const dual &x, const dual &y) {
return {x.val * y.val, x.val * y.eps + x.eps * y.val};
}
dual &operator+=(const dual &x) { return *this = *this + x; }
dual &operator-=(const dual &x) { return *this = *this - x; }
dual &operator*=(const T &x) { return *this = *this * x; }
dual &operator/=(const T &x) { return *this = *this / x; }
dual &operator*=(const dual &x) { return *this = *this * x; }
friend constexpr bool operator==(const dual &x, const dual &y) {
return x.val == y.val;
};
friend constexpr bool operator<(const dual &x, const dual &y) {
return x.val < y.val;
};
friend constexpr bool operator!=(const dual &x, const dual &y) {
return !(x == y);
}
friend constexpr bool operator>(const dual &x, const dual &y) {
return y < x;
}
friend constexpr bool operator<=(const dual &x, const dual &y) {
return !(x > y);
}
friend constexpr bool operator>=(const dual &x, const dual &y) {
return !(x < y);
}
friend ostream &operator<<(ostream &os, const dual &x) {
return os << x.val << "+eps*" << x.val;
}
};
} // namespace Maxwell
namespace std {
template <typename T> using dual = Maxwell::dual<T>;
template <typename T> struct equal_to<dual<T> > {
constexpr bool operator()(const dual<T> &x, const dual<T> &y) const {
return equal_to<T>()(x.val, y.val) && equal_to<T>()(x.eps, y.eps);
}
};
template <typename T> struct less<dual<T> > {
constexpr bool operator()(const dual<T> &x, const dual<T> &y) const {
return less<T>(x.val, y.val) ||
(equal_to<T>(x.val, y.val) && less<T>(x.eps, y.eps));
}
};
template <typename T> constexpr dual<T> cos(const dual<T> &x);
template <typename T> constexpr dual<T> exp(const dual<T> &x);
template <typename T> constexpr dual<T> sin(const dual<T> &x);
template <typename T> constexpr dual<T> sqrt(const dual<T> &x);
template <typename T> constexpr dual<T> cos(const dual<T> &x) {
return {cos(x.val), -sin(x.val) * x.eps};
}
template <typename T> constexpr dual<T> exp(const dual<T> &x) {
auto r = exp(x.val);
return {r, r * x.eps};
}
template <typename T> constexpr dual<T> sin(const dual<T> &x) {
return {sin(x.val), cos(x.val) * x.eps};
}
template <typename T> constexpr dual<T> sqrt(const dual<T> &x) {
auto r = sqrt(x.val);
return {r, x.eps / (2 * r)};
}
} // namespace std
namespace {
template <typename T, int CI, int CJ, int CK>
T average(const Loop::GF3D<const T, CI, CJ, CK> &var,
const Loop::PointDesc &p) {
const auto DI = Loop::vect<int, Loop::dim>::unit(0);
const auto DJ = Loop::vect<int, Loop::dim>::unit(1);
const auto DK = Loop::vect<int, Loop::dim>::unit(2);
T res = 0;
for (int k = 0; k < 2 - CK; ++k)
for (int j = 0; j < 2 - CJ; ++j)
for (int i = 0; i < 2 - CI; ++i)
res += var(p.I + i * DI + j * DJ + k * DK);
return res / ((2 - CI) * (2 - CJ) * (2 - CK));
}
} // namespace
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ex_(cctkGH, ex);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ey_(cctkGH, ey);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> ez_(cctkGH, ez);
const Loop::GF3D<const CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const Loop::GF3D<const CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const Loop::GF3D<const CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const Loop::GF3D<const CCTK_REAL, 0, 1, 1> curlayz_(cctkGH, curlayz);
const Loop::GF3D<const CCTK_REAL, 1, 0, 1> curlazx_(cctkGH, curlazx);
const Loop::GF3D<const CCTK_REAL, 1, 1, 0> curlaxy_(cctkGH, curlaxy);
const Loop::GF3D<const CCTK_REAL, 0, 0, 0> dive_(cctkGH, dive);
const Loop::GF3D<const CCTK_REAL, 1, 1, 1> divb_(cctkGH, divb);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgphi_(cctkGH, avgphi);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgax_(cctkGH, avgax);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgay_(cctkGH, avgay);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgaz_(cctkGH, avgaz);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgex_(cctkGH, avgex);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgey_(cctkGH, avgey);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgez_(cctkGH, avgez);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgbyz_(cctkGH, avgbyz);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgbzx_(cctkGH, avgbzx);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgbxy_(cctkGH, avgbxy);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgcurlayz_(cctkGH, avgcurlyz);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgcurlazx_(cctkGH, avgcurlzx);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgcurlaxy_(cctkGH, avgcurlxy);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> avgdive_(cctkGH, avgdive);
const auto DI = vect<int, dim>::unit(0);
const auto DJ = vect<int, dim>::unit(1);
const auto DK = vect<int, dim>::unit(2);
Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
avgphi_(p.I) = average(phi_, p);
});
const GF3D<const CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const GF3D<const CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const GF3D<const CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgax_(p.I) = average(ax_, p); });
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgay_(p.I) = average(ay_, p); });
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgaz_(p.I) = average(az_, p); });
const GF3D<CCTK_REAL, 1, 1, 1> avgdyz_(cctkGH, avgdyz);
const GF3D<CCTK_REAL, 1, 1, 1> avgdzx_(cctkGH, avgdzx);
const GF3D<CCTK_REAL, 1, 1, 1> avgdxy_(cctkGH, avgdxy);
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgex_(p.I) = average(ex_, p); });
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgey_(p.I) = average(ey_, p); });
Loop::loop_int<1, 1, 1>(
cctkGH, [&](const Loop::PointDesc &p) { avgez_(p.I) = average(ez_, p); });
const GF3D<CCTK_REAL, 1, 1, 1> avgbyz_(cctkGH, avgbyz);
const GF3D<CCTK_REAL, 1, 1, 1> avgbzx_(cctkGH, avgbzx);
const GF3D<CCTK_REAL, 1, 1, 1> avgbxy_(cctkGH, avgbxy);
Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
avgcurlayz_(p.I) = average(curlayz_, p);
});
Loop::loop_int<1, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
avgcurlazx_(p.I) = average(curlazx_, p);
loop_int<1, 1, 1>(cctkGH, [&](const PointDesc &p) {
avgbyz_(p.I) = (byz_(p.I) + byz_(p.I + DI)) / 2;
const auto DI = Loop::vect<int, Loop::dim>::unit(0);
const auto DJ = Loop::vect<int, Loop::dim>::unit(1);
const auto DK = Loop::vect<int, Loop::dim>::unit(2);
const auto DI = vect<int, dim>::unit(0);
const auto DJ = vect<int, dim>::unit(1);
const auto DK = vect<int, dim>::unit(2);
const auto dxm{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DI)) / dx;
}};
const auto dym{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DJ)) / dy;
}};
const auto dzm{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DK)) / dz;
}};
const auto dxp{
[&](const auto &u, const auto &I) { return (u(I + DI) - u(I)) / dx; }};
const auto dyp{
[&](const auto &u, const auto &I) { return (u(I + DJ) - u(I)) / dy; }};
const auto dzp{
[&](const auto &u, const auto &I) { return (u(I + DK) - u(I)) / dz; }};
const auto dxp{[&](const auto &u, const auto &p) {
return (u(p.I + DI) - u(p.I)) / dx;
}};
const auto dyp{[&](const auto &u, const auto &p) {
return (u(p.I + DJ) - u(p.I)) / dy;
}};
const auto dzp{[&](const auto &u, const auto &p) {
return (u(p.I + DK) - u(p.I)) / dz;
}};
const GF3D<const CCTK_REAL, 0, 1, 1> dyz_(cctkGH, dyz);
const GF3D<const CCTK_REAL, 1, 0, 1> dzx_(cctkGH, dzx);
const GF3D<const CCTK_REAL, 1, 1, 0> dxy_(cctkGH, dxy);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const GF3D<const CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const GF3D<const CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const GF3D<const CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const Loop::GF3D<CCTK_REAL, 0, 1, 1> curlayz_(cctkGH, curlayz);
const Loop::GF3D<CCTK_REAL, 1, 0, 1> curlazx_(cctkGH, curlazx);
const Loop::GF3D<CCTK_REAL, 1, 1, 0> curlaxy_(cctkGH, curlaxy);
const Loop::GF3D<CCTK_REAL, 0, 0, 0> dive_(cctkGH, dive);
const Loop::GF3D<CCTK_REAL, 1, 1, 1> divb_(cctkGH, divb);
Loop::loop_int<0, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
curlayz_(p.I) = byz_(p.I) - (dyp(az_, p) - dzp(ay_, p));
});
Loop::loop_int<1, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
curlazx_(p.I) = bzx_(p.I) - (dzp(ax_, p) - dxp(az_, p));
});
Loop::loop_int<1, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
curlaxy_(p.I) = bxy_(p.I) - (dxp(ay_, p) - dyp(ax_, p));
loop_int<1, 1, 1>(cctkGH, [&](const PointDesc &p) {
divd_(p.I) = dxp(dyz_, p.I) + dyp(dzx_, p.I) + dzp(dxy_, p.I);
////////////////////////////////////////////////////////////////////////////////
template <typename T> struct potential {
// Electric scalar potential
T phi;
// Electric vector potential (to ensure div E = 0)
T cyz, czx, cxy;
// Magnetic vector potential
T ax, ay, az;
};
// Continuous derivative
template <typename F, typename T>
potential<T> calc_dt(const F &f, T t, T x, T y, T z) {
auto fd = f(dual<T>(t, 1), dual<T>(x), dual<T>(y), dual<T>(z));
return {
fd.phi.eps, fd.cyz.eps, fd.czx.eps, fd.cxy.eps,
fd.ax.eps, fd.ay.eps, fd.az.eps,
};
}
// Discrete derivatives (centred)
template <typename F, typename T>
potential<T> calc_dxc(const F &f, T t, T x, T y, T z, T dx) {
auto fm = f(t, x - dx / 2, y, z);
auto fp = f(t, x + dx / 2, y, z);
return {
.phi = (fp.phi - fm.phi) / dx,
.cyz = (fp.cyz - fm.cyz) / dx,
.czx = (fp.czx - fm.czx) / dx,
.cxy = (fp.cxy - fm.cxy) / dx,
.ax = (fp.ax - fm.ax) / dx,
.ay = (fp.ay - fm.ay) / dx,
.az = (fp.az - fm.az) / dx,
};
}
template <typename F, typename T>
potential<T> calc_dyc(const F &f, T t, T x, T y, T z, T dy) {
auto fm = f(t, x, y - dy / 2, z);
auto fp = f(t, x, y + dy / 2, z);
return {
.phi = (fp.phi - fm.phi) / dy,
.cyz = (fp.cyz - fm.cyz) / dy,
.czx = (fp.czx - fm.czx) / dy,
.cxy = (fp.cxy - fm.cxy) / dy,
.ax = (fp.ax - fm.ax) / dy,
.ay = (fp.ay - fm.ay) / dy,
.az = (fp.az - fm.az) / dy,
};
}
template <typename F, typename T>
potential<T> calc_dzc(const F &f, T t, T x, T y, T z, T dz) {
auto fm = f(t, x, y, z - dz / 2);
auto fp = f(t, x, y, z + dz / 2);
return {
.phi = (fp.phi - fm.phi) / dz,
.cyz = (fp.cyz - fm.cyz) / dz,
.czx = (fp.czx - fm.czx) / dz,
.cxy = (fp.cxy - fm.cxy) / dz,
.ax = (fp.ax - fm.ax) / dz,
.ay = (fp.ay - fm.ay) / dz,
.az = (fp.az - fm.az) / dz,
};
}
////////////////////////////////////////////////////////////////////////////////
// Plane wave
template <typename T>
potential<T> plane_wave(const T t, const T x, const T y, const T z) {
DECLARE_CCTK_PARAMETERS;
// wave number
T kx = M_PI * spatial_frequency_x;
T ky = M_PI * spatial_frequency_y;
T kz = M_PI * spatial_frequency_z;
assert(kx == 0);
assert(ky == 0);
T omega = sqrt(pow2(kx) + pow2(ky) + pow2(kz));
// amplitude
T hx = amplitude_x;
T hy = amplitude_y;
T hz = amplitude_z;
assert(hy == 0);
assert(hz == 0);
// solution
assert(t == 0);
T u = sin(omega * t - kz * z);
return {
.phi = 0,
.cyz = 0,
.czx = hx / kz * u,
.cxy = 0,
.ax = -hx / kz * u,
.ay = 0,
.az = 0,
};
}
const CCTK_REAL dx = CCTK_DELTA_SPACE(0);
const CCTK_REAL dy = CCTK_DELTA_SPACE(1);
const CCTK_REAL dz = CCTK_DELTA_SPACE(2);
const Loop::GF3D<CCTK_REAL, 0, 0, 0> phi_(cctkGH, phi);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> ex_(cctkGH, ex);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> ey_(cctkGH, ey);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> ez_(cctkGH, ez);
const Loop::GF3D<CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const Loop::GF3D<CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const Loop::GF3D<CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const GF3D<CCTK_REAL, 0, 1, 1> dyz_(cctkGH, dyz);
const GF3D<CCTK_REAL, 1, 0, 1> dzx_(cctkGH, dzx);
const GF3D<CCTK_REAL, 1, 1, 0> dxy_(cctkGH, dxy);
const auto loop_setup{[&](const auto &f4) {
const auto f{[&](const auto &p) { return f4(t, p.x, p.y, p.z); }};
const auto dtf{[&](const auto &p) {
return calc_dt(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z);
}};
const auto dxf{[&](const auto &p) {
return calc_dxc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dx);
}};
const auto dyf{[&](const auto &p) {
return calc_dyc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dy);
}};
const auto dzf{[&](const auto &p) {
return calc_dzc(
[&](auto t, auto x, auto y, auto z) { return f4(t, x, y, z); }, t,
p.x, p.y, p.z, dz);
}};
const GF3D<CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const GF3D<CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const GF3D<CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
Loop::loop_int<1, 0, 0>(
cctkGH, [&](const Loop::PointDesc &p) { ax_(p.I) = f(p).ax; });
Loop::loop_int<0, 1, 0>(
cctkGH, [&](const Loop::PointDesc &p) { ay_(p.I) = f(p).ay; });
Loop::loop_int<0, 0, 1>(
cctkGH, [&](const Loop::PointDesc &p) { az_(p.I) = f(p).az; });
// wave number
const CCTK_REAL kx = CCTK_REAL(M_PI) * spatial_frequency_x;
const CCTK_REAL ky = CCTK_REAL(M_PI) * spatial_frequency_y;
const CCTK_REAL kz = CCTK_REAL(M_PI) * spatial_frequency_z;
const CCTK_REAL omega = sqrt(pow2(kx) + pow2(ky) + pow2(kz));
// amplitude
const CCTK_REAL hx = amplitude_x;
const CCTK_REAL hy = amplitude_y;
const CCTK_REAL hz = amplitude_z;
Loop::loop_int<1, 0, 0>(cctkGH, [&](const Loop::PointDesc &p) {
ex_(p.I) = -dxf(p).phi + dyf(p).cxy - dzf(p).czx;
loop_int<0, 1, 1>(cctkGH, [&](const PointDesc &p) {
dyz_(p.I) = hx * cos(omega * t - kx * p.x - ky * p.y - kz * p.z);
Loop::loop_int<0, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
ey_(p.I) = -dyf(p).phi + dzf(p).cyz - dxf(p).cxy;
loop_int<1, 0, 1>(cctkGH, [&](const PointDesc &p) {
dzx_(p.I) = hy * cos(omega * t - kx * p.x - ky * p.y - kz * p.z);
Loop::loop_int<0, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
ez_(p.I) = -dzf(p).phi + dxf(p).czx - dyf(p).cyz;
loop_int<1, 1, 0>(cctkGH, [&](const PointDesc &p) {
dxy_(p.I) = hz * cos(omega * t - kx * p.x - ky * p.y - kz * p.z);
using namespace Loop;
using namespace std;
namespace {
template <int CI, int CJ, int CK, typename T>
T star(const GF3D<const T, CI, CJ, CK> &u, const vect<int, dim> &I) {
constexpr int SI = CI == 0 ? +1 : -1;
constexpr int SJ = CJ == 0 ? +1 : -1;
constexpr int SK = CK == 0 ? +1 : -1;
const auto DI = vect<int, dim>::unit(0);
const auto DJ = vect<int, dim>::unit(1);
const auto DK = vect<int, dim>::unit(2);
const auto SDI = SI * DI;
const auto SDJ = SJ * DJ;
const auto SDK = SK * DK;
return (u(I) + u(I + SDI) + u(I + SDJ) + u(I + SDI + SDJ) + u(I + SDK) +
u(I + SDI + SDK) + u(I + SDJ + SDK) + u(I + SDI + SDJ + SDK)) /
8;
}
const auto DI = Loop::vect<int, Loop::dim>::unit(0);
const auto DJ = Loop::vect<int, Loop::dim>::unit(1);
const auto DK = Loop::vect<int, Loop::dim>::unit(2);
const auto DI = vect<int, dim>::unit(0);
const auto DJ = vect<int, dim>::unit(1);
const auto DK = vect<int, dim>::unit(2);
const auto dxm{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DI)) / dx;
}};
const auto dym{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DJ)) / dy;
}};
const auto dzm{[&](const auto &u, const auto &p) {
return (u(p.I) - u(p.I - DK)) / dz;
}};
const auto dxp{
[&](const auto &u, const auto &I) { return (u(I + DI) - u(I)) / dx; }};
const auto dyp{
[&](const auto &u, const auto &I) { return (u(I + DJ) - u(I)) / dy; }};
const auto dzp{
[&](const auto &u, const auto &I) { return (u(I + DK) - u(I)) / dz; }};
const auto dxp{[&](const auto &u, const auto &p) {
return (u(p.I + DI) - u(p.I)) / dx;
}};
const auto dyp{[&](const auto &u, const auto &p) {
return (u(p.I + DJ) - u(p.I)) / dy;
}};
const auto dzp{[&](const auto &u, const auto &p) {
return (u(p.I + DK) - u(p.I)) / dz;
}};
const GF3D<const CCTK_REAL, 0, 1, 1> dyz_(cctkGH, dyz);
const GF3D<const CCTK_REAL, 1, 0, 1> dzx_(cctkGH, dzx);
const GF3D<const CCTK_REAL, 1, 1, 0> dxy_(cctkGH, dxy);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ax_(cctkGH, ax);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ay_(cctkGH, ay);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> az_(cctkGH, az);
const GF3D<CCTK_REAL, 0, 1, 1> dtdyz_(cctkGH, dtdyz);
const GF3D<CCTK_REAL, 1, 0, 1> dtdzx_(cctkGH, dtdzx);
const GF3D<CCTK_REAL, 1, 1, 0> dtdxy_(cctkGH, dtdxy);
const Loop::GF3D<const CCTK_REAL, 1, 0, 0> ex_(cctkGH, ex);
const Loop::GF3D<const CCTK_REAL, 0, 1, 0> ey_(cctkGH, ey);
const Loop::GF3D<const CCTK_REAL, 0, 0, 1> ez_(cctkGH, ez);
const Loop::GF3D<const CCTK_REAL, 0, 1, 1> byz_(cctkGH, byz);
const Loop::GF3D<const CCTK_REAL, 1, 0, 1> bzx_(cctkGH, bzx);
const Loop::GF3D<const CCTK_REAL, 1, 1, 0> bxy_(cctkGH, bxy);
const Loop::GF3D<CCTK_REAL, 0, 0, 0> dtphi_(cctkGH, dtphi);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> dtax_(cctkGH, dtax);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> dtay_(cctkGH, dtay);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> dtaz_(cctkGH, dtaz);
const Loop::GF3D<CCTK_REAL, 1, 0, 0> dtex_(cctkGH, dtex);
const Loop::GF3D<CCTK_REAL, 0, 1, 0> dtey_(cctkGH, dtey);
const Loop::GF3D<CCTK_REAL, 0, 0, 1> dtez_(cctkGH, dtez);
const GF3D<CCTK_REAL, 0, 1, 1> dtbyz_(cctkGH, dtbyz);
const GF3D<CCTK_REAL, 1, 0, 1> dtbzx_(cctkGH, dtbzx);
const GF3D<CCTK_REAL, 1, 1, 0> dtbxy_(cctkGH, dtbxy);
const Loop::GF3D<CCTK_REAL, 0, 1, 1> dtbyz_(cctkGH, dtbyz);
const Loop::GF3D<CCTK_REAL, 1, 0, 1> dtbzx_(cctkGH, dtbzx);
const Loop::GF3D<CCTK_REAL, 1, 1, 0> dtbxy_(cctkGH, dtbxy);
Loop::loop_int<0, 0, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtphi_(p.I) = -(dxm(ax_, p) + dym(ay_, p) + dzm(az_, p));
loop_int<0, 1, 1>(cctkGH, [&](const PointDesc &p) {
dtdyz_(p.I) = (star(bxy_, p.I + DJ) - star(bxy_, p.I)) / dy -
(star(bzx_, p.I + DK) - star(bzx_, p.I)) / dz;
Loop::loop_int<1, 0, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtax_(p.I) = -dxp(phi_, p) - ex_(p.I);
loop_int<1, 0, 1>(cctkGH, [&](const PointDesc &p) {
dtdzx_(p.I) = (star(byz_, p.I + DK) - star(byz_, p.I)) / dz -
(star(bxy_, p.I + DI) - star(bxy_, p.I)) / dx;
Loop::loop_int<0, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtay_(p.I) = -dyp(phi_, p) - ey_(p.I);
loop_int<1, 1, 0>(cctkGH, [&](const PointDesc &p) {
dtdxy_(p.I) = (star(bzx_, p.I + DI) - star(bzx_, p.I)) / dx -
(star(byz_, p.I + DJ) - star(byz_, p.I)) / dy;
Loop::loop_int<1, 0, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtex_(p.I) = -(dzm(bzx_, p) - dym(bxy_, p));
loop_int<0, 1, 1>(cctkGH, [&](const PointDesc &p) {
dtbyz_(p.I) = (star(dzx_, p.I + DK) - star(dzx_, p.I)) / dz -
(star(dxy_, p.I + DJ) - star(dxy_, p.I)) / dy;
Loop::loop_int<0, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtey_(p.I) = -(dxm(bxy_, p) - dzm(byz_, p));
});
Loop::loop_int<0, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
dtez_(p.I) = -(dym(byz_, p) - dxm(bzx_, p));
});
Loop::loop_int<0, 1, 1>(cctkGH, [&](const Loop::PointDesc &p) {
dtbyz_(p.i, p.j, p.k) = dzp(ey_, p) - dyp(ez_, p);
});
Loop::loop_int<1, 0, 1>(cctkGH, [&](const Loop::PointDesc &p) {
dtbzx_(p.i, p.j, p.k) = dxp(ez_, p) - dzp(ex_, p);
loop_int<1, 0, 1>(cctkGH, [&](const PointDesc &p) {
dtbzx_(p.I) = (star(dxy_, p.I + DI) - star(dxy_, p.I)) / dx -
(star(dyz_, p.I + DK) - star(dyz_, p.I)) / dz;
Loop::loop_int<1, 1, 0>(cctkGH, [&](const Loop::PointDesc &p) {
dtbxy_(p.i, p.j, p.k) = dyp(ex_, p) - dxp(ey_, p);
loop_int<1, 1, 0>(cctkGH, [&](const PointDesc &p) {
dtbxy_(p.I) = (star(dyz_, p.I + DJ) - star(dyz_, p.I)) / dy -
(star(dzx_, p.I + DI) - star(dzx_, p.I)) / dx;