def declare_BSSN_gridfunctions_if_not_declared_already():
# Step 2: Register all needed BSSN gridfunctions.
# Declare as globals all variables that may be
# used outside this function
global hDD, aDD, lambdaU, vetU, betU, trK, cf, alpha
# Check to see if this function has already been called.
# If so, do not register the gridfunctions again!
for i in range(len(gri.glb_gridfcs_list)):
if "hDD00" in gri.glb_gridfcs_list[i].name:
hDD = ixp.declarerank2("hDD", "sym01")
aDD = ixp.declarerank2("aDD", "sym01")
lambdaU = ixp.declarerank1("lambdaU")
vetU = ixp.declarerank1("vetU")
betU = ixp.declarerank1("betU")
trK, cf, alpha = sp.symbols('trK cf alpha', real=True)
return hDD, aDD, lambdaU, vetU, betU, trK, cf, alpha
# Step 2.a: Register indexed quantities, using ixp.register_... functions
hDD = ixp.register_gridfunctions_for_single_rank2("EVOL", "hDD", "sym01")
aDD = ixp.register_gridfunctions_for_single_rank2("EVOL", "aDD", "sym01")
lambdaU = ixp.register_gridfunctions_for_single_rank1("EVOL", "lambdaU")
vetU = ixp.register_gridfunctions_for_single_rank1("EVOL", "vetU")
betU = ixp.register_gridfunctions_for_single_rank1("EVOL", "betU")
# Step 2.b: Register scalar quantities, using gri.register_gridfunctions()
trK, cf, alpha = gri.register_gridfunctions("EVOL", ["trK", "cf", "alpha"])
return hDD, aDD, lambdaU, vetU, betU, trK, cf, alpha
def generate_everything_for_UnitTesting():
# First define hydrodynamical quantities
u4U = ixp.declarerank1("u4U", DIM=4)
rho_b, P, epsilon = sp.symbols('rho_b P epsilon', real=True)
# Then ADM quantities
gammaDD = ixp.declarerank2("gammaDD", "sym01", DIM=3)
KDD = ixp.declarerank2("KDD", "sym01", DIM=3)
betaU = ixp.declarerank1("betaU", DIM=3)
alpha = sp.symbols('alpha', real=True)
# First compute stress-energy tensor T4UU and T4UD:
compute_T4UU(gammaDD, betaU, alpha, rho_b, P, epsilon, u4U)
compute_T4UD(gammaDD, betaU, alpha, T4UU)
# Next sqrt(gamma)
compute_sqrtgammaDET(gammaDD)
# Compute conservative variables in terms of primitive variables
compute_rho_star(alpha, sqrtgammaDET, rho_b, u4U)
compute_tau_tilde(alpha, sqrtgammaDET, T4UU, rho_star)
compute_S_tildeD(alpha, sqrtgammaDET, T4UD)
# Then compute v^i from u^mu
compute_vU_from_u4U__no_speed_limit(u4U)
# Next compute fluxes of conservative variables
compute_rho_star_fluxU(vU, rho_star)
compute_tau_tilde_fluxU(alpha, sqrtgammaDET, vU, T4UU, rho_star)
compute_S_tilde_fluxUD(alpha, sqrtgammaDET, T4UD)
# Then declare derivatives & compute g4DD_zerotimederiv_dD
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01", DIM=3)
betaU_dD = ixp.declarerank2("betaU_dD", "nosym", DIM=3)
alpha_dD = ixp.declarerank1("alpha_dD", DIM=3)
compute_g4DD_zerotimederiv_dD(gammaDD, betaU, alpha, gammaDD_dD, betaU_dD,
alpha_dD)
# Then compute source terms on tau_tilde and S_tilde equations
compute_s_source_term(KDD, betaU, alpha, sqrtgammaDET, alpha_dD, T4UU)
compute_S_tilde_source_termD(alpha, sqrtgammaDET, g4DD_zerotimederiv_dD,
T4UU)
# Then compute the 4-velocities in terms of an input Valencia 3-velocity testValenciavU[i]
testValenciavU = ixp.declarerank1("testValenciavU", DIM=3)
u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(
alpha, betaU, gammaDD, testValenciavU)
# Finally compute the 4-velocities in terms of an input 3-velocity testvU[i] = u^i/u^0
testvU = ixp.declarerank1("testvU", DIM=3)
u4U_in_terms_of_vU__rescale_vU_by_applying_speed_limit(
alpha, betaU, gammaDD, testvU)
def ScalarField_Tmunu():
global T4UU
# Step 1.c: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.d: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 1.e: Import all basic (unrescaled) BSSN scalars & tensors
Bq.BSSN_basic_tensors()
alpha = Bq.alpha
betaU = Bq.betaU
# Step 1.g: Define ADM quantities in terms of BSSN quantities
BtoA.ADM_in_terms_of_BSSN()
gammaDD = BtoA.gammaDD
gammaUU = BtoA.gammaUU
# Step 1.h: Define scalar field quantitites
sf_dD = ixp.declarerank1("sf_dD")
Pi = sp.Symbol("sfM", real=True)
# Step 2a: Set up \partial^{t}\varphi = Pi/alpha
sf4dU = ixp.zerorank1(DIM=4)
sf4dU[0] = Pi / alpha
# Step 2b: Set up \partial^{i}\varphi = -Pi*beta^{i}/alpha + gamma^{ij}\partial_{j}\varphi
for i in range(DIM):
sf4dU[i + 1] = -Pi * betaU[i] / alpha
for j in range(DIM):
sf4dU[i + 1] += gammaUU[i][j] * sf_dD[j]
# Step 2c: Set up \partial^{i}\varphi\partial_{i}\varphi = -Pi**2 + gamma^{ij}\partial_{i}\varphi\partial_{j}\varphi
sf4d2 = -Pi**2
for i in range(DIM):
for j in range(DIM):
sf4d2 += gammaUU[i][j] * sf_dD[i] * sf_dD[j]
# Step 3a: Setting up g^{\mu\nu}
ADMg.g4UU_ito_BSSN_or_ADM("ADM",
gammaDD=gammaDD,
betaU=betaU,
alpha=alpha,
gammaUU=gammaUU)
g4UU = ADMg.g4UU
# Step 3b: Setting up T^{\mu\nu} for a massless scalar field
T4UU = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
T4UU[mu][nu] = sf4dU[mu] * sf4dU[nu] - g4UU[mu][nu] * sf4d2 / 2
def GiRaFFE_NRPy_Afield_flux(Ccodesdir):
cmd.mkdir(Ccodesdir)
# Write out the code to a file.
gammaDD = ixp.declarerank2("gammaDD", "sym01", DIM=3)
betaU = ixp.declarerank1("betaU", DIM=3)
alpha = sp.sympify("alpha")
for flux_dirn in range(3):
chsp.find_cmax_cmin(flux_dirn, gammaDD, betaU, alpha)
Ccode_kernel = outputC([chsp.cmax, chsp.cmin], ["cmax", "cmin"],
"returnstring",
params="outCverbose=False,CSE_sorting=none")
Ccode_kernel = Ccode_kernel.replace("cmax",
"*cmax").replace("cmin", "*cmin")
Ccode_kernel = Ccode_kernel.replace("betaU0", "betaUi").replace(
"betaU1", "betaUi").replace("betaU2", "betaUi")
with open(
os.path.join(Ccodesdir,
"compute_cmax_cmin_dirn" + str(flux_dirn) + ".h"),
"w") as file:
file.write(Ccode_kernel)
with open(os.path.join(Ccodesdir, name + ".h"), "w") as file:
file.write(body)
def write_out_functions_for_StildeD_source_term(outdir,outCparams,gammaDD,betaU,alpha,ValenciavU,BU,sqrt4pi):
generate_memory_access_code()
# First, we declare some dummy tensors that we will use for the codegen.
gammaDDdD = ixp.declarerank3("gammaDDdD","sym01",DIM=3)
betaUdD = ixp.declarerank2("betaUdD","nosym",DIM=3)
alphadD = ixp.declarerank1("alphadD",DIM=3)
# We need to rerun a few of these functions with the reset lists to make sure these functions
# don't cheat by using analytic expressions
GRHD.compute_sqrtgammaDET(gammaDD)
GRHD.u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha, betaU, gammaDD, ValenciavU)
GRFFE.compute_smallb4U(gammaDD, betaU, alpha, GRHD.u4U_ito_ValenciavU, BU, sqrt4pi)
GRFFE.compute_smallbsquared(gammaDD, betaU, alpha, GRFFE.smallb4U)
GRFFE.compute_TEM4UU(gammaDD,betaU,alpha, GRFFE.smallb4U, GRFFE.smallbsquared,GRHD.u4U_ito_ValenciavU)
GRHD.compute_g4DD_zerotimederiv_dD(gammaDD,betaU,alpha, gammaDDdD,betaUdD,alphadD)
GRHD.compute_S_tilde_source_termD(alpha, GRHD.sqrtgammaDET,GRHD.g4DD_zerotimederiv_dD, GRFFE.TEM4UU)
for i in range(3):
desc = "Adds the source term to StildeD"+str(i)+"."
name = "calculate_StildeD"+str(i)+"_source_term"
outCfunction(
outfile = os.path.join(outdir,name+".h"), desc=desc, name=name,
params ="const paramstruct *params,const REAL *auxevol_gfs, REAL *rhs_gfs",
body = general_access \
+metric_deriv_access[i]\
+outputC(GRHD.S_tilde_source_termD[i],"Stilde_rhsD"+str(i),"returnstring",params=outCparams).replace("IDX4","IDX4S")\
+write_final_quantity[i],
loopopts ="InteriorPoints",
rel_path_for_Cparams=os.path.join("../"))
def BSSN_source_terms_for_BSSN_constraints(custom_T4UU=None):
global sourceterm_H, sourceterm_MU
# Step 4.a: Call BSSN_source_terms_ito_T4UU to get SDD, SD, S, & rho
if custom_T4UU == "unrescaled BSSN source terms already given":
SDD = ixp.declarerank2("SDD", "sym01")
SD = ixp.declarerank1("SD")
S = sp.symbols("S", real=True)
rho = sp.symbols("rho", real=True)
else:
SDD, SD, S, rho = stress_energy_source_terms_ito_T4UU_and_ADM_or_BSSN_metricvars(
"BSSN", custom_T4UU)
PI = par.Cparameters("REAL", thismodule, ["PI"],
"3.14159265358979323846264338327950288")
# Step 4.b: Add source term to the Hamiltonian constraint H
sourceterm_H = -16 * PI * rho
# Step 4.c: Add source term to the momentum constraint M^i
# Step 4.c.i: Compute gammaUU in terms of BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as AitoB
AitoB.ADM_in_terms_of_BSSN() # Provides gammaUU
# Step 4.c.ii: Raise S_i
SU = ixp.zerorank1()
for i in range(3):
for j in range(3):
SU[i] += AitoB.gammaUU[i][j] * SD[j]
# Step 4.c.iii: Add source term to momentum constraint & rescale:
sourceterm_MU = ixp.zerorank1()
for i in range(3):
sourceterm_MU[i] = -8 * PI * SU[i] / rfm.ReU[i]
def generate_everything_for_UnitTesting():
# First define hydrodynamical quantities
u4U = ixp.declarerank1("u4U", DIM=4)
B_tildeU = ixp.declarerank1("B_tildeU", DIM=3)
# Then ADM quantities
gammaDD = ixp.declarerank2("gammaDD", "sym01", DIM=3)
betaU = ixp.declarerank1("betaU", DIM=3)
alpha = sp.symbols('alpha', real=True)
# Then numerical constant
sqrt4pi = sp.symbols('sqrt4pi', real=True)
# First compute stress-energy tensor T4UU and T4UD:
import GRHD.equations as GHeq
GHeq.compute_sqrtgammaDET(gammaDD)
compute_B_notildeU(GHeq.sqrtgammaDET, B_tildeU)
compute_smallb4U(gammaDD, betaU, alpha, u4U, B_notildeU, sqrt4pi)
compute_smallb4U_with_driftvU_for_FFE(gammaDD, betaU, alpha, u4U,
B_notildeU, sqrt4pi)
compute_smallbsquared(gammaDD, betaU, alpha, smallb4U)
compute_TEM4UU(gammaDD, betaU, alpha, smallb4U, smallbsquared, u4U)
compute_TEM4UD(gammaDD, betaU, alpha, TEM4UU)
# Compute conservative variables in terms of primitive variables
GHeq.compute_S_tildeD(alpha, GHeq.sqrtgammaDET, TEM4UD)
global S_tildeD
S_tildeD = GHeq.S_tildeD
# Next compute fluxes of conservative variables
GHeq.compute_S_tilde_fluxUD(alpha, GHeq.sqrtgammaDET, TEM4UD)
global S_tilde_fluxUD
S_tilde_fluxUD = GHeq.S_tilde_fluxUD
# Then declare derivatives & compute g4DDdD
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01", DIM=3)
betaU_dD = ixp.declarerank2("betaU_dD", "nosym", DIM=3)
alpha_dD = ixp.declarerank1("alpha_dD", DIM=3)
GHeq.compute_g4DD_zerotimederiv_dD(gammaDD, betaU, alpha, gammaDD_dD,
betaU_dD, alpha_dD)
# Finally compute source terms on tau_tilde and S_tilde equations
GHeq.compute_S_tilde_source_termD(alpha, GHeq.sqrtgammaDET,
GHeq.g4DD_zerotimederiv_dD, TEM4UU)
global S_tilde_source_termD
S_tilde_source_termD = GHeq.S_tilde_source_termD
def betU_vetU(betaU, BU):
global vetU, betU
if betaU == None:
betaU = ixp.declarerank1("betaU")
if BU == None:
BU = ixp.declarerank1("BU")
if rfm.have_already_called_reference_metric_function == False:
print(
"BSSN.BSSN_in_terms_of_ADM.bet_vet(): Must call reference_metric() first!"
)
sys.exit(1)
vetU = ixp.zerorank1()
betU = ixp.zerorank1()
for i in range(DIM):
vetU[i] = betaU[i] / rfm.ReU[i]
betU[i] = BU[i] / rfm.ReU[i]
def ScalarWaveCurvilinear_RHSs():
# Step 1: Get the spatial dimension, defined in the
# NRPy+ "grid" module. With reference metrics,
# this must be set to 3 or fewer.
DIM = par.parval_from_str("DIM")
# Step 2: Set up the reference metric and
# quantities derived from the
# reference metric.
rfm.reference_metric()
# Step 3: Compute the contracted Christoffel symbols:
contractedGammahatU = ixp.zerorank1()
for k in range(DIM):
for i in range(DIM):
for j in range(DIM):
contractedGammahatU[k] += rfm.ghatUU[i][j] * rfm.GammahatUDD[k][i][j]
# Step 4: Register gridfunctions that are needed as input
# to the scalar wave RHS expressions.
uu, vv = gri.register_gridfunctions("EVOL",["uu","vv"])
# Step 5a: Declare the rank-1 indexed expression \partial_{i} u,
# Derivative variables like these must have an underscore
# in them, so the finite difference module can parse the
# variable name properly.
uu_dD = ixp.declarerank1("uu_dD")
# Step 5b: Declare the rank-2 indexed expression \partial_{ij} u,
# which is symmetric about interchange of indices i and j
# Derivative variables like these must have an underscore
# in them, so the finite difference module can parse the
# variable name properly.
uu_dDD = ixp.declarerank2("uu_dDD","sym01")
# Step 7: Specify RHSs as global variables,
# to enable access outside this
# function (e.g., for C code output)
global uu_rhs,vv_rhs
# Step 6: Define right-hand sides for the evolution.
# Step 6a: uu_rhs = vv:
uu_rhs = vv
# Step 6b: The right-hand side of the \partial_t v equation
# is given by:
# \hat{g}^{ij} \partial_i \partial_j u - \hat{\Gamma}^i \partial_i u.
# ^^^^^^^^^^^^ PART 1 ^^^^^^^^^^^^^^^^ ^^^^^^^^^^ PART 2 ^^^^^^^^^^^
vv_rhs = 0
for i in range(DIM):
# PART 2:
vv_rhs -= contractedGammahatU[i]*uu_dD[i]
for j in range(DIM):
# PART 1:
vv_rhs += rfm.ghatUU[i][j]*uu_dDD[i][j]
vv_rhs *= wavespeed*wavespeed
def BSSN_source_terms_for_BSSN_RHSs(custom_T4UU=None):
global sourceterm_trK_rhs, sourceterm_a_rhsDD, sourceterm_lambda_rhsU, sourceterm_Lambdabar_rhsU
# Step 3.a: Call BSSN_source_terms_ito_T4UU to get SDD, SD, S, & rho
if custom_T4UU == "unrescaled BSSN source terms already given":
SDD = ixp.declarerank2("SDD", "sym01")
SD = ixp.declarerank1("SD")
S = sp.symbols("S", real=True)
rho = sp.symbols("rho", real=True)
else:
SDD, SD, S, rho = stress_energy_source_terms_ito_T4UU_and_ADM_or_BSSN_metricvars(
"BSSN", custom_T4UU)
PI = par.Cparameters("REAL", thismodule, ["PI"],
"3.14159265358979323846264338327950288")
alpha = sp.symbols("alpha", real=True)
# Step 3.b: trK_rhs
sourceterm_trK_rhs = 4 * PI * alpha * (rho + S)
# Step 3.c: Abar_rhsDD:
# Step 3.c.i: Compute trace-free part of S_{ij}:
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors() # Sets gammabarDD
gammabarUU, dummydet = ixp.symm_matrix_inverter3x3(
Bq.gammabarDD) # Set gammabarUU
tracefree_SDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
tracefree_SDD[i][j] = SDD[i][j]
for i in range(3):
for j in range(3):
for k in range(3):
for m in range(3):
tracefree_SDD[i][j] += -sp.Rational(1, 3) * Bq.gammabarDD[
i][j] * gammabarUU[k][m] * SDD[k][m]
# Step 3.c.ii: Define exp_m4phi = e^{-4 phi}
Bq.phi_and_derivs()
# Step 3.c.iii: Evaluate stress-energy part of AbarDD's RHS
sourceterm_a_rhsDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
Abar_rhsDDij = -8 * PI * alpha * Bq.exp_m4phi * tracefree_SDD[i][j]
sourceterm_a_rhsDD[i][j] = Abar_rhsDDij / rfm.ReDD[i][j]
# Step 3.d: Stress-energy part of Lambdabar_rhsU = stressenergy_Lambdabar_rhsU
sourceterm_Lambdabar_rhsU = ixp.zerorank1()
for i in range(3):
for j in range(3):
sourceterm_Lambdabar_rhsU[
i] += -16 * PI * alpha * gammabarUU[i][j] * SD[j]
sourceterm_lambda_rhsU = ixp.zerorank1()
for i in range(3):
sourceterm_lambda_rhsU[i] = sourceterm_Lambdabar_rhsU[i] / rfm.ReU[i]
def BrillLindquist(ComputeADMGlobalsOnly=False,
include_NRPy_basic_defines_and_pickle=False):
# Step 2: Setting up Brill-Lindquist initial data
# Step 2.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
global Cartxyz, gammaCartDD, KCartDD, alphaCart, betaCartU, BCartU
Cartxyz = ixp.declarerank1("Cartxyz")
# Step 2.b: Set psi, the conformal factor:
psi = sp.sympify(1)
psi += BH1_mass / (2 * sp.sqrt((Cartxyz[0] - BH1_posn_x)**2 +
(Cartxyz[1] - BH1_posn_y)**2 +
(Cartxyz[2] - BH1_posn_z)**2))
psi += BH2_mass / (2 * sp.sqrt((Cartxyz[0] - BH2_posn_x)**2 +
(Cartxyz[1] - BH2_posn_y)**2 +
(Cartxyz[2] - BH2_posn_z)**2))
# Step 2.c: Set all needed ADM variables in Cartesian coordinates
gammaCartDD = ixp.zerorank2()
KCartDD = ixp.zerorank2() # K_{ij} = 0 for these initial data
for i in range(DIM):
gammaCartDD[i][i] = psi**4
alphaCart = 1 / psi**2
betaCartU = ixp.zerorank1(
) # We generally choose \beta^i = 0 for these initial data
BCartU = ixp.zerorank1(
) # We generally choose B^i = 0 for these initial data
if ComputeADMGlobalsOnly == True:
return
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Cartesian",Cartxyz,
gammaCartDD,KCartDD,alphaCart,betaCartU,BCartU)
import BSSN.BSSN_ID_function_string as bIDf
# Generates initial_data() C function & stores to outC_function_dict["initial_data"]
bIDf.BSSN_ID_function_string(
cf,
hDD,
lambdaU,
aDD,
trK,
alpha,
vetU,
betU,
include_NRPy_basic_defines=include_NRPy_basic_defines_and_pickle)
if include_NRPy_basic_defines_and_pickle:
return pickle_NRPy_env()
def BrillLindquist(ComputeADMGlobalsOnly=False):
global Cartxyz, gammaCartDD, KCartDD, alphaCart, betaCartU, BCartU
# Step 2: Setting up Brill-Lindquist initial data
thismodule = "Brill-Lindquist"
BH1_posn_x, BH1_posn_y, BH1_posn_z = par.Cparameters(
"REAL", thismodule, ["BH1_posn_x", "BH1_posn_y", "BH1_posn_z"])
BH1_mass = par.Cparameters("REAL", thismodule, ["BH1_mass"])
BH2_posn_x, BH2_posn_y, BH2_posn_z = par.Cparameters(
"REAL", thismodule, ["BH2_posn_x", "BH2_posn_y", "BH2_posn_z"])
BH2_mass = par.Cparameters("REAL", thismodule, ["BH2_mass"])
# Step 2.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
global Cartxyz, gammaCartDD, KCartDD, alphaCart, betaCartU, BCartU
Cartxyz = ixp.declarerank1("Cartxyz")
# Step 2.b: Set psi, the conformal factor:
psi = sp.sympify(1)
psi += BH1_mass / (2 * sp.sqrt((Cartxyz[0] - BH1_posn_x)**2 +
(Cartxyz[1] - BH1_posn_y)**2 +
(Cartxyz[2] - BH1_posn_z)**2))
psi += BH2_mass / (2 * sp.sqrt((Cartxyz[0] - BH2_posn_x)**2 +
(Cartxyz[1] - BH2_posn_y)**2 +
(Cartxyz[2] - BH2_posn_z)**2))
# Step 2.c: Set all needed ADM variables in Cartesian coordinates
gammaCartDD = ixp.zerorank2()
KCartDD = ixp.zerorank2() # K_{ij} = 0 for these initial data
for i in range(DIM):
gammaCartDD[i][i] = psi**4
alphaCart = 1 / psi**2
betaCartU = ixp.zerorank1(
) # We generally choose \beta^i = 0 for these initial data
BCartU = ixp.zerorank1(
) # We generally choose B^i = 0 for these initial data
if ComputeADMGlobalsOnly == True:
return
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Cartesian",Cartxyz,
gammaCartDD,KCartDD,alphaCart,betaCartU,BCartU)
global returnfunction
returnfunction = bIDf.BSSN_ID_function_string(cf, hDD, lambdaU, aDD, trK,
alpha, vetU, betU)
def calculate_E_i_flux(flux_dirn,inputs_provided=True,alpha_face=None,gamma_faceDD=None,beta_faceU=None,\
Valenciav_rU=None,B_rU=None,Valenciav_lU=None,B_lU=None):
if not inputs_provided:
# declare all variables
alpha_face = sp.symbols(alpha_face)
beta_faceU = ixp.declarerank1("beta_faceU")
gamma_faceDD = ixp.declarerank2("gamma_faceDD", "sym01")
Valenciav_rU = ixp.declarerank1("Valenciav_rU")
B_rU = ixp.declarerank1("B_rU")
Valenciav_lU = ixp.declarerank1("Valenciav_lU")
B_lU = ixp.declarerank1("B_lU")
global E_fluxD
E_fluxD = ixp.zerorank1()
for field_comp in range(3):
find_cmax_cmin(field_comp, gamma_faceDD, beta_faceU, alpha_face)
calculate_flux_and_state_for_Induction(field_comp,flux_dirn, gamma_faceDD,beta_faceU,alpha_face,\
Valenciav_rU,B_rU)
Fr = F
Ur = U
calculate_flux_and_state_for_Induction(field_comp,flux_dirn, gamma_faceDD,beta_faceU,alpha_face,\
Valenciav_lU,B_lU)
Fl = F
Ul = U
E_fluxD[field_comp] += HLLE_solver(cmax, cmin, Fr, Fl, Ur, Ul)
def setup_ADM_quantities(inputvars):
if inputvars == "ADM":
gammaDD = ixp.declarerank2("gammaDD", "sym01")
betaU = ixp.declarerank1("betaU")
alpha = sp.symbols("alpha", real=True)
elif inputvars == "BSSN":
import BSSN.ADM_in_terms_of_BSSN as AitoB
# Construct gamma_{ij} in terms of cf & gammabar_{ij}
AitoB.ADM_in_terms_of_BSSN()
gammaDD = AitoB.gammaDD
# Next construct beta^i in terms of vet^i and reference metric quantities
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors()
betaU = Bq.betaU
alpha = sp.symbols("alpha", real=True)
else:
print("inputvars = " + str(inputvars) + " not supported. Please choose ADM or BSSN.")
sys.exit(1)
return gammaDD,betaU,alpha
def add_to_Cfunction_dict__AD_gauge_term_psi6Phi_fin_diff(includes=None):
xi_damping = par.Cparameters("REAL",thismodule,"xi_damping",0.1)
GRFFE.compute_psi6Phi_rhs_damping_term(alpha,psi6Phi,xi_damping)
AevolParen_dD = ixp.declarerank1("AevolParen_dD",DIM=3)
PhievolParenU_dD = ixp.declarerank2("PhievolParenU_dD","nosym",DIM=3)
A_rhsD = ixp.zerorank1()
psi6Phi_rhs = GRFFE.psi6Phi_damping
for i in range(3):
A_rhsD[i] += -AevolParen_dD[i]
psi6Phi_rhs += -PhievolParenU_dD[i][i]
# Add Kreiss-Oliger dissipation to the GRFFE RHSs:
# psi6Phi_dKOD = ixp.declarerank1("psi6Phi_dKOD")
# AD_dKOD = ixp.declarerank2("AD_dKOD","nosym")
# for i in range(3):
# psi6Phi_rhs += diss_strength*psi6Phi_dKOD[i]*rfm.ReU[i] # ReU[i] = 1/scalefactor_orthog_funcform[i]
# for j in range(3):
# A_rhsD[j] += diss_strength*AD_dKOD[j][i]*rfm.ReU[i] # ReU[i] = 1/scalefactor_orthog_funcform[i]
RHSs_to_print = [
lhrh(lhs=gri.gfaccess("rhs_gfs","AD0"),rhs=A_rhsD[0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","AD1"),rhs=A_rhsD[1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","AD2"),rhs=A_rhsD[2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","psi6Phi"),rhs=psi6Phi_rhs),
]
desc = "Calculate AD gauge term and psi6Phi RHSs"
name = "calculate_AD_gauge_psi6Phi_RHSs"
params ="const paramstruct *params,const REAL *in_gfs,const REAL *auxevol_gfs,REAL *rhs_gfs"
body = fin.FD_outputC("returnstring",RHSs_to_print,params=outCparams)
loopopts ="InteriorPoints"
add_to_Cfunction_dict(
includes=includes,
desc=desc,
name=name, params=params,
body=body, loopopts=loopopts)
outC_function_dict[name] = outC_function_dict[name].replace("= NGHOSTS","= NGHOSTS_A2B").replace("NGHOSTS+Nxx0","Nxx_plus_2NGHOSTS0-NGHOSTS_A2B").replace("NGHOSTS+Nxx1","Nxx_plus_2NGHOSTS1-NGHOSTS_A2B").replace("NGHOSTS+Nxx2","Nxx_plus_2NGHOSTS2-NGHOSTS_A2B")
def GiRaFFE_NRPy_P2C(gammaDD, betaU, alpha, ValenciavU, BU, sqrt4pi):
# After recalculating the 3-velocity, we need to update the poynting flux:
# We'll reset the Valencia velocity, since this will be part of a second call to outCfunction.
ValenciavU = ixp.declarerank1("ValenciavU", DIM=3)
# First compute stress-energy tensor T4UU and T4UD:
GRHD.compute_sqrtgammaDET(gammaDD)
GRHD.u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(
alpha, betaU, gammaDD, ValenciavU)
GRFFE.compute_smallb4U(gammaDD, betaU, alpha, GRHD.u4U_ito_ValenciavU, BU,
sqrt4pi)
GRFFE.compute_smallbsquared(gammaDD, betaU, alpha, GRFFE.smallb4U)
GRFFE.compute_TEM4UU(gammaDD, betaU, alpha, GRFFE.smallb4U,
GRFFE.smallbsquared, GRHD.u4U_ito_ValenciavU)
GRFFE.compute_TEM4UD(gammaDD, betaU, alpha, GRFFE.TEM4UU)
# Compute conservative variables in terms of primitive variables
GRHD.compute_S_tildeD(alpha, GRHD.sqrtgammaDET, GRFFE.TEM4UD)
global StildeD
StildeD = GRHD.S_tildeD
def phi_and_derivs():
# Step 9.a: Declare as globals all expressions that may be used
# outside this function, declare BSSN gridfunctions
# if not defined already, and set DIM=3.
global phi_dD, phi_dupD, phi_dDD, exp_m4phi, phi_dBarD, phi_dBarDD
hDD, aDD, lambdaU, vetU, betU, trK, cf, alpha = declare_BSSN_gridfunctions_if_not_declared_already(
)
DIM = 3
# GammabarUDD is used below, defined in
# gammabar__inverse_and_derivs()
gammabar__inverse_and_derivs()
# Step 9.a.i: Define partial derivatives of \phi in terms of evolved quantity "cf":
cf_dD = ixp.declarerank1("cf_dD")
cf_dupD = ixp.declarerank1("cf_dupD") # Needed for \partial_t \phi next.
cf_dDD = ixp.declarerank2("cf_dDD", "sym01")
phi_dD = ixp.zerorank1()
phi_dupD = ixp.zerorank1()
phi_dDD = ixp.zerorank2()
exp_m4phi = sp.sympify(0)
# Step 9.a.ii: Assuming cf=phi, define exp_m4phi, phi_dD,
# phi_dupD (upwind finite-difference version of phi_dD), and phi_DD
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "phi":
for i in range(DIM):
phi_dD[i] = cf_dD[i]
phi_dupD[i] = cf_dupD[i]
for j in range(DIM):
phi_dDD[i][j] = cf_dDD[i][j]
exp_m4phi = sp.exp(-4 * cf)
# Step 9.a.iii: Assuming cf=W=e^{-2 phi}, define exp_m4phi, phi_dD,
# phi_dupD (upwind finite-difference version of phi_dD), and phi_DD
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "W":
# \partial_i W = \partial_i (e^{-2 phi}) = -2 e^{-2 phi} \partial_i phi
# -> \partial_i phi = -\partial_i cf / (2 cf)
for i in range(DIM):
phi_dD[i] = -cf_dD[i] / (2 * cf)
phi_dupD[i] = -cf_dupD[i] / (2 * cf)
for j in range(DIM):
# \partial_j \partial_i phi = - \partial_j [\partial_i cf / (2 cf)]
# = - cf_{,ij} / (2 cf) + \partial_i cf \partial_j cf / (2 cf^2)
phi_dDD[i][j] = (-cf_dDD[i][j] +
cf_dD[i] * cf_dD[j] / cf) / (2 * cf)
exp_m4phi = cf * cf
# Step 9.a.iv: Assuming cf=chi=e^{-4 phi}, define exp_m4phi, phi_dD,
# phi_dupD (upwind finite-difference version of phi_dD), and phi_DD
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "chi":
# \partial_i chi = \partial_i (e^{-4 phi}) = -4 e^{-4 phi} \partial_i phi
# -> \partial_i phi = -\partial_i cf / (4 cf)
for i in range(DIM):
phi_dD[i] = -cf_dD[i] / (4 * cf)
phi_dupD[i] = -cf_dupD[i] / (4 * cf)
for j in range(DIM):
# \partial_j \partial_i phi = - \partial_j [\partial_i cf / (4 cf)]
# = - cf_{,ij} / (4 cf) + \partial_i cf \partial_j cf / (4 cf^2)
phi_dDD[i][j] = (-cf_dDD[i][j] +
cf_dD[i] * cf_dD[j] / cf) / (4 * cf)
exp_m4phi = cf
# Step 9.a.v: Error out if unsupported EvolvedConformalFactor_cf choice is made:
cf_choice = par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf")
if not (cf_choice == "phi" or cf_choice == "W" or cf_choice == "chi"):
print("Error: EvolvedConformalFactor_cf == " + par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") +
" unsupported!")
exit(1)
# Step 9.b: Define phi_dBarD = phi_dD (since phi is a scalar) and phi_dBarDD (covariant derivative)
# \bar{D}_i \bar{D}_j \phi = \phi_{;\bar{i}\bar{j}} = \bar{D}_i \phi_{,j}
# = \phi_{,ij} - \bar{\Gamma}^k_{ij} \phi_{,k}
phi_dBarD = phi_dD
phi_dBarDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
phi_dBarDD[i][j] = phi_dDD[i][j]
for k in range(DIM):
phi_dBarDD[i][j] += -GammabarUDD[k][i][j] * phi_dD[k]
def BSSN_RHSs():
# Step 1.c: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
global have_already_called_BSSN_RHSs_function # setting to global enables other modules to see updated value.
have_already_called_BSSN_RHSs_function = True
# Step 1.d: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.e: Import all basic (unrescaled) BSSN scalars & tensors
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors()
gammabarDD = Bq.gammabarDD
AbarDD = Bq.AbarDD
LambdabarU = Bq.LambdabarU
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
# Step 1.f: Import all neeeded rescaled BSSN tensors:
aDD = Bq.aDD
cf = Bq.cf
lambdaU = Bq.lambdaU
# Step 2.a.i: Import derivative expressions for betaU defined in the BSSN.BSSN_quantities module:
Bq.betaU_derivs()
betaU_dD = Bq.betaU_dD
betaU_dDD = Bq.betaU_dDD
# Step 2.a.ii: Import derivative expression for gammabarDD
Bq.gammabar__inverse_and_derivs()
gammabarDD_dupD = Bq.gammabarDD_dupD
# Step 2.a.iii: First term of \partial_t \bar{\gamma}_{i j} right-hand side:
# \beta^k \bar{\gamma}_{ij,k} + \beta^k_{,i} \bar{\gamma}_{kj} + \beta^k_{,j} \bar{\gamma}_{ik}
gammabar_rhsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
gammabar_rhsDD[i][j] += betaU[k] * gammabarDD_dupD[i][j][k] + betaU_dD[k][i] * gammabarDD[k][j] \
+ betaU_dD[k][j] * gammabarDD[i][k]
# Step 2.b.i: First import \bar{A}_{ij} = AbarDD[i][j], and its contraction trAbar = \bar{A}^k_k
# from BSSN.BSSN_quantities
Bq.AbarUU_AbarUD_trAbar_AbarDD_dD()
trAbar = Bq.trAbar
# Step 2.b.ii: Import detgammabar quantities from BSSN.BSSN_quantities:
Bq.detgammabar_and_derivs()
detgammabar = Bq.detgammabar
detgammabar_dD = Bq.detgammabar_dD
# Step 2.b.ii: Compute the contraction \bar{D}_k \beta^k = \beta^k_{,k} + \frac{\beta^k \bar{\gamma}_{,k}}{2 \bar{\gamma}}
Dbarbetacontraction = sp.sympify(0)
for k in range(DIM):
Dbarbetacontraction += betaU_dD[k][
k] + betaU[k] * detgammabar_dD[k] / (2 * detgammabar)
# Step 2.b.iii: Second term of \partial_t \bar{\gamma}_{i j} right-hand side:
# \frac{2}{3} \bar{\gamma}_{i j} \left (\alpha \bar{A}_{k}^{k} - \bar{D}_{k} \beta^{k}\right )
for i in range(DIM):
for j in range(DIM):
gammabar_rhsDD[i][j] += sp.Rational(2, 3) * gammabarDD[i][j] * (
alpha * trAbar - Dbarbetacontraction)
# Step 2.c: Third term of \partial_t \bar{\gamma}_{i j} right-hand side:
# -2 \alpha \bar{A}_{ij}
for i in range(DIM):
for j in range(DIM):
gammabar_rhsDD[i][j] += -2 * alpha * AbarDD[i][j]
# Step 3.a: First term of \partial_t \bar{A}_{i j}:
# \beta^k \partial_k \bar{A}_{ij} + \partial_i \beta^k \bar{A}_{kj} + \partial_j \beta^k \bar{A}_{ik}
# First define AbarDD_dupD:
AbarDD_dupD = Bq.AbarDD_dupD # From Bq.AbarUU_AbarUD_trAbar_AbarDD_dD()
Abar_rhsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Abar_rhsDD[i][j] += betaU[k] * AbarDD_dupD[i][j][k] + betaU_dD[k][i] * AbarDD[k][j] \
+ betaU_dD[k][j] * AbarDD[i][k]
# Step 3.b: Second term of \partial_t \bar{A}_{i j}:
# - (2/3) \bar{A}_{i j} \bar{D}_{k} \beta^{k} - 2 \alpha \bar{A}_{i k} {\bar{A}^{k}}_{j} + \alpha \bar{A}_{i j} K
gammabarUU = Bq.gammabarUU # From Bq.gammabar__inverse_and_derivs()
AbarUD = Bq.AbarUD # From Bq.AbarUU_AbarUD_trAbar()
for i in range(DIM):
for j in range(DIM):
Abar_rhsDD[i][j] += -sp.Rational(2, 3) * AbarDD[i][
j] * Dbarbetacontraction + alpha * AbarDD[i][j] * trK
for k in range(DIM):
Abar_rhsDD[i][j] += -2 * alpha * AbarDD[i][k] * AbarUD[k][j]
# Step 3.c.i: Define partial derivatives of \phi in terms of evolved quantity "cf":
Bq.phi_and_derivs()
phi_dD = Bq.phi_dD
phi_dupD = Bq.phi_dupD
phi_dDD = Bq.phi_dDD
exp_m4phi = Bq.exp_m4phi
phi_dBarD = Bq.phi_dBarD # phi_dBarD = Dbar_i phi = phi_dD (since phi is a scalar)
phi_dBarDD = Bq.phi_dBarDD # phi_dBarDD = Dbar_i Dbar_j phi (covariant derivative)
# Step 3.c.ii: Define RbarDD
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
RbarDD = Bq.RbarDD
# Step 3.c.iii: Define first and second derivatives of \alpha, as well as
# \bar{D}_i \bar{D}_j \alpha, which is defined just like phi
alpha_dD = ixp.declarerank1("alpha_dD")
alpha_dDD = ixp.declarerank2("alpha_dDD", "sym01")
alpha_dBarD = alpha_dD
alpha_dBarDD = ixp.zerorank2()
GammabarUDD = Bq.GammabarUDD # Defined in Bq.gammabar__inverse_and_derivs()
for i in range(DIM):
for j in range(DIM):
alpha_dBarDD[i][j] = alpha_dDD[i][j]
for k in range(DIM):
alpha_dBarDD[i][j] += -GammabarUDD[k][i][j] * alpha_dD[k]
# Step 3.c.iv: Define the terms in curly braces:
curlybrackettermsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
curlybrackettermsDD[i][j] = -2 * alpha * phi_dBarDD[i][j] + 4 * alpha * phi_dBarD[i] * phi_dBarD[j] \
+ 2 * alpha_dBarD[i] * phi_dBarD[j] \
+ 2 * alpha_dBarD[j] * phi_dBarD[i] \
- alpha_dBarDD[i][j] + alpha * RbarDD[i][j]
# Step 3.c.v: Compute the trace:
curlybracketterms_trace = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
curlybracketterms_trace += gammabarUU[i][j] * curlybrackettermsDD[
i][j]
# Step 3.c.vi: Third and final term of Abar_rhsDD[i][j]:
for i in range(DIM):
for j in range(DIM):
Abar_rhsDD[i][j] += exp_m4phi * (
curlybrackettermsDD[i][j] -
sp.Rational(1, 3) * gammabarDD[i][j] * curlybracketterms_trace)
# Step 4: Right-hand side of conformal factor variable "cf". Supported
# options include: cf=phi, cf=W=e^(-2*phi) (default), and cf=chi=e^(-4*phi)
# \partial_t phi = \left[\beta^k \partial_k \phi \right] <- TERM 1
# + \frac{1}{6} \left (\bar{D}_{k} \beta^{k} - \alpha K \right ) <- TERM 2
global cf_rhs
cf_rhs = sp.Rational(1, 6) * (Dbarbetacontraction - alpha * trK) # Term 2
for k in range(DIM):
cf_rhs += betaU[k] * phi_dupD[k] # Term 1
# Next multiply to convert phi_rhs to cf_rhs.
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "phi":
pass # do nothing; cf_rhs = phi_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "W":
cf_rhs *= -2 * cf # cf_rhs = -2*cf*phi_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "chi":
cf_rhs *= -4 * cf # cf_rhs = -4*cf*phi_rhs
else:
print("Error: EvolvedConformalFactor_cf == " + par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") +
" unsupported!")
exit(1)
# Step 5: right-hand side of trK (trace of extrinsic curvature):
# \partial_t K = \beta^k \partial_k K <- TERM 1
# + \frac{1}{3} \alpha K^{2} <- TERM 2
# + \alpha \bar{A}_{i j} \bar{A}^{i j} <- TERM 3
# - - e^{-4 \phi} (\bar{D}_{i} \bar{D}^{i} \alpha + 2 \bar{D}^{i} \alpha \bar{D}_{i} \phi ) <- TERM 4
global trK_rhs
# TERM 2:
trK_rhs = sp.Rational(1, 3) * alpha * trK * trK
trK_dupD = ixp.declarerank1("trK_dupD")
for i in range(DIM):
# TERM 1:
trK_rhs += betaU[i] * trK_dupD[i]
for i in range(DIM):
for j in range(DIM):
# TERM 4:
trK_rhs += -exp_m4phi * gammabarUU[i][j] * (
alpha_dBarDD[i][j] + 2 * alpha_dBarD[j] * phi_dBarD[i])
AbarUU = Bq.AbarUU # From Bq.AbarUU_AbarUD_trAbar()
for i in range(DIM):
for j in range(DIM):
# TERM 3:
trK_rhs += alpha * AbarDD[i][j] * AbarUU[i][j]
# Step 6: right-hand side of \partial_t \bar{\Lambda}^i:
# \partial_t \bar{\Lambda}^i = \beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k <- TERM 1
# + \bar{\gamma}^{j k} \hat{D}_{j} \hat{D}_{k} \beta^{i} <- TERM 2
# + \frac{2}{3} \Delta^{i} \bar{D}_{j} \beta^{j} <- TERM 3
# + \frac{1}{3} \bar{D}^{i} \bar{D}_{j} \beta^{j} <- TERM 4
# - 2 \bar{A}^{i j} (\partial_{j} \alpha - 6 \partial_{j} \phi) <- TERM 5
# + 2 \alpha \bar{A}^{j k} \Delta_{j k}^{i} <- TERM 6
# - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_{j} K <- TERM 7
# Step 6.a: Term 1 of \partial_t \bar{\Lambda}^i: \beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k
# First we declare \bar{\Lambda}^i and \bar{\Lambda}^i_{,j} in terms of \lambda^i and \lambda^i_{,j}
global LambdabarU_dupD # Used on the RHS of the Gamma-driving shift conditions
LambdabarU_dupD = ixp.zerorank2()
lambdaU_dupD = ixp.declarerank2("lambdaU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
LambdabarU_dupD[i][j] = lambdaU_dupD[i][j] * rfm.ReU[i] + lambdaU[
i] * rfm.ReUdD[i][j]
global Lambdabar_rhsU # Used on the RHS of the Gamma-driving shift conditions
Lambdabar_rhsU = ixp.zerorank1()
for i in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[i] += betaU[k] * LambdabarU_dupD[i][k] - betaU_dD[
i][k] * LambdabarU[k] # Term 1
# Step 6.b: Term 2 of \partial_t \bar{\Lambda}^i = \bar{\gamma}^{jk} (Term 2a + Term 2b + Term 2c)
# Term 2a: \bar{\gamma}^{jk} \beta^i_{,kj}
Term2aUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Term2aUDD[i][j][k] += betaU_dDD[i][k][j]
# Term 2b: \hat{\Gamma}^i_{mk,j} \beta^m + \hat{\Gamma}^i_{mk} \beta^m_{,j}
# + \hat{\Gamma}^i_{dj}\beta^d_{,k} - \hat{\Gamma}^d_{kj} \beta^i_{,d}
Term2bUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
Term2bUDD[i][j][k] += rfm.GammahatUDDdD[i][m][k][j] * betaU[m] \
+ rfm.GammahatUDD[i][m][k] * betaU_dD[m][j] \
+ rfm.GammahatUDD[i][m][j] * betaU_dD[m][k] \
- rfm.GammahatUDD[m][k][j] * betaU_dD[i][m]
# Term 2c: \hat{\Gamma}^i_{dj}\hat{\Gamma}^d_{mk} \beta^m - \hat{\Gamma}^d_{kj} \hat{\Gamma}^i_{md} \beta^m
Term2cUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
for d in range(DIM):
Term2cUDD[i][j][k] += (rfm.GammahatUDD[i][d][j] * rfm.GammahatUDD[d][m][k] \
- rfm.GammahatUDD[d][k][j] * rfm.GammahatUDD[i][m][d]) * betaU[m]
Lambdabar_rhsUpieceU = ixp.zerorank1()
# Put it all together to get Term 2:
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[i] += gammabarUU[j][k] * (Term2aUDD[i][j][k] +
Term2bUDD[i][j][k] +
Term2cUDD[i][j][k])
Lambdabar_rhsUpieceU[i] += gammabarUU[j][k] * (
Term2aUDD[i][j][k] + Term2bUDD[i][j][k] +
Term2cUDD[i][j][k])
# Step 6.c: Term 3 of \partial_t \bar{\Lambda}^i:
# \frac{2}{3} \Delta^{i} \bar{D}_{j} \beta^{j}
DGammaU = Bq.DGammaU # From Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
for i in range(DIM):
Lambdabar_rhsU[i] += sp.Rational(
2, 3) * DGammaU[i] * Dbarbetacontraction # Term 3
# Step 6.d: Term 4 of \partial_t \bar{\Lambda}^i:
# \frac{1}{3} \bar{D}^{i} \bar{D}_{j} \beta^{j}
detgammabar_dDD = Bq.detgammabar_dDD # From Bq.detgammabar_and_derivs()
Dbarbetacontraction_dBarD = ixp.zerorank1()
for k in range(DIM):
for m in range(DIM):
Dbarbetacontraction_dBarD[m] += betaU_dDD[k][k][m] + \
(betaU_dD[k][m] * detgammabar_dD[k] +
betaU[k] * detgammabar_dDD[k][m]) / (2 * detgammabar) \
- betaU[k] * detgammabar_dD[k] * detgammabar_dD[m] / (
2 * detgammabar * detgammabar)
for i in range(DIM):
for m in range(DIM):
Lambdabar_rhsU[i] += sp.Rational(
1, 3) * gammabarUU[i][m] * Dbarbetacontraction_dBarD[m]
# Step 6.e: Term 5 of \partial_t \bar{\Lambda}^i:
# - 2 \bar{A}^{i j} (\partial_{j} \alpha - 6 \alpha \partial_{j} \phi)
for i in range(DIM):
for j in range(DIM):
Lambdabar_rhsU[i] += -2 * AbarUU[i][j] * (alpha_dD[j] -
6 * alpha * phi_dD[j])
# Step 6.f: Term 6 of \partial_t \bar{\Lambda}^i:
# 2 \alpha \bar{A}^{j k} \Delta^{i}_{j k}
DGammaUDD = Bq.DGammaUDD # From RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[
i] += 2 * alpha * AbarUU[j][k] * DGammaUDD[i][j][k]
# Step 6.g: Term 7 of \partial_t \bar{\Lambda}^i:
# -\frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_{j} K
trK_dD = ixp.declarerank1("trK_dD")
for i in range(DIM):
for j in range(DIM):
Lambdabar_rhsU[i] += -sp.Rational(
4, 3) * alpha * gammabarUU[i][j] * trK_dD[j]
# Step 7: Rescale the RHS quantities so that the evolved
# variables are smooth across coord singularities
global h_rhsDD, a_rhsDD, lambda_rhsU
h_rhsDD = ixp.zerorank2()
a_rhsDD = ixp.zerorank2()
lambda_rhsU = ixp.zerorank1()
for i in range(DIM):
lambda_rhsU[i] = Lambdabar_rhsU[i] / rfm.ReU[i]
for j in range(DIM):
h_rhsDD[i][j] = gammabar_rhsDD[i][j] / rfm.ReDD[i][j]
a_rhsDD[i][j] = Abar_rhsDD[i][j] / rfm.ReDD[i][j]
def MaxwellCartesian_Evol():
#Step 0: Set the spatial dimension parameter to 3.
par.set_parval_from_str("grid::DIM", 3)
DIM = par.parval_from_str("grid::DIM")
# Step 1: Set the finite differencing order to 4.
# (not needed here)
# par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", 4)
# Step 2: Register gridfunctions that are needed as input.
_psi = gri.register_gridfunctions(
"EVOL", ["psi"]) # lgtm [py/unused-local-variable]
# Step 3a: Declare the rank-1 indexed expressions E_{i}, A_{i},
# and \partial_{i} \psi. Derivative variables like these
# must have an underscore in them, so the finite
# difference module can parse the variable name properly.
ED = ixp.register_gridfunctions_for_single_rank1("EVOL", "ED")
AD = ixp.register_gridfunctions_for_single_rank1("EVOL", "AD")
psi_dD = ixp.declarerank1("psi_dD")
## Step 3b: Declare the conformal metric tensor and its first
# derivative. These are needed to find the Christoffel
# symbols, which we need for covariant derivatives.
gammaDD = ixp.register_gridfunctions_for_single_rank2(
"AUX", "gammaDD", "sym01") # The AUX or EVOL designation is *not*
# used in diagnostic modules.
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01")
gammaDD_dDD = ixp.declarerank4("gammaDD_dDD", "sym01_sym23")
gammaUU, detgamma = ixp.symm_matrix_inverter3x3(gammaDD)
gammaUU_dD = ixp.declarerank3("gammaDD_dD", "sym01")
# Define the Christoffel symbols
GammaUDD = ixp.zerorank3(DIM)
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
GammaUDD[i][k][l] += (sp.Rational(1,2))*gammaUU[i][m]*\
(gammaDD_dD[m][k][l] + gammaDD_dD[m][l][k] - gammaDD_dD[k][l][m])
# Step 3b: Declare the rank-2 indexed expression \partial_{j} A_{i},
# which is not symmetric in its indices.
# Derivative variables like these must have an underscore
# in them, so the finite difference module can parse the
# variable name properly.
AD_dD = ixp.declarerank2("AD_dD", "nosym")
# Step 3c: Declare the rank-3 indexed expression \partial_{jk} A_{i},
# which is symmetric in the two {jk} indices.
AD_dDD = ixp.declarerank3("AD_dDD", "sym12")
# Step 4: Calculate first and second covariant derivatives, and the
# necessary contractions.
# First covariant derivative
# D_{j} A_{i} = A_{i,j} - \Gamma^{k}_{ij} A_{k}
AD_dcovD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
AD_dcovD[i][j] = AD_dD[i][j]
for k in range(DIM):
AD_dcovD[i][j] -= GammaUDD[k][i][j] * AD[k]
# First, we must construct the lowered Christoffel symbols:
# \Gamma_{ijk} = \gamma_{il} \Gamma^l_{jk}
# And raise the index on A:
# A^j = \gamma^{ij} A_i
GammaDDD = ixp.zerorank3()
AU = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
AU[j] += gammaUU[i][j] * AD[i]
for k in range(DIM):
for l in range(DIM):
GammaDDD[i][j][k] += gammaDD[i][l] * GammaUDD[l][j][k]
# Covariant second derivative (the bracketed terms):
# D_j D^j A_i = \gamma^{jk} [A_{i,jk} - A^l (\gamma_{li,kj} + \gamma_{kl,ij} - \gamma_{ik,lj})
# + \Gamma_{lik} (\gamma^{lm} A_{m,j} + A_m \gamma^{lm}{}_{,j})
# - (\Gamma^l_{ij} A_{l,k} + \Gamma^l_{jk} A_{i,l})
# + (\Gamma^m_{ij} \Gamma^l_{mk} A_l + \Gamma ^m_{jk} \Gamma^l_{im} A_l)
AD_dcovDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
AD_dcovDD[i][j][k] = AD_dDD[i][j][k]
for l in range(DIM):
# Terms 1 and 3
AD_dcovDD[i][j][k] -= AU[l] * (gammaDD_dDD[l][i][k][j] + gammaDD_dDD[k][l][i][j] - \
gammaDD_dDD[i][k][l][j]) \
+ GammaUDD[l][i][j] * AD_dD[l][k] + GammaUDD[l][j][k] * AD_dD[i][l]
for m in range(DIM):
# Terms 2 and 4
AD_dcovDD[i][j][k] += GammaDDD[l][i][k] * (gammaUU[l][m] * AD_dD[m][j] + AD[m] * gammaUU_dD[l][m][j]) \
+ GammaUDD[m][i][j] * GammaUDD[l][m][k] * AD[l] \
+ GammaUDD[m][j][k] * GammaUDD[l][i][m] * AD[l]
# Covariant divergence
# D_{i} A^{i} = \gamma^{ij} D_{j} A_{i}
DivA = 0
# Gradient of covariant divergence
# DivA_dD_{i} = \gamma^{jk} A_{k;\hat{j}\hat{i}}
DivA_dD = ixp.zerorank1()
# Covariant Laplacian
# LapAD_{i} = \gamma^{jk} A_{i;\hat{j}\hat{k}}
LapAD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
DivA += gammaUU[i][j] * AD_dcovD[i][j]
for k in range(DIM):
DivA_dD[i] += gammaUU[j][k] * AD_dcovDD[k][j][i]
LapAD[i] += gammaUU[j][k] * AD_dcovDD[i][j][k]
global ArhsD, ErhsD, psi_rhs
system = par.parval_from_str("System_to_use")
if system == "System_I":
# Step 5: Define right-hand sides for the evolution.
print("Warning: System I is less stable!")
ArhsD = ixp.zerorank1()
ErhsD = ixp.zerorank1()
for i in range(DIM):
ArhsD[i] = -ED[i] - psi_dD[i]
ErhsD[i] = -LapAD[i] + DivA_dD[i]
psi_rhs = -DivA
elif system == "System_II":
# We inherit here all of the definitions from System I, above
# Step 7a: Register the scalar auxiliary variable \Gamma
Gamma = gri.register_gridfunctions("EVOL", ["Gamma"])
# Step 7b: Declare the ordinary gradient \partial_{i} \Gamma
Gamma_dD = ixp.declarerank1("Gamma_dD")
# Step 8a: Construct the second covariant derivative of the scalar \psi
# \psi_{;\hat{i}\hat{j}} = \psi_{,i;\hat{j}}
# = \psi_{,ij} - \Gamma^{k}_{ij} \psi_{,k}
psi_dDD = ixp.declarerank2("psi_dDD", "sym01")
psi_dcovDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
psi_dcovDD[i][j] = psi_dDD[i][j]
for k in range(DIM):
psi_dcovDD[i][j] += -GammaUDD[k][i][j] * psi_dD[k]
# Step 8b: Construct the covariant Laplacian of \psi
# Lappsi = ghat^{ij} D_{j} D_{i} \psi
Lappsi = 0
for i in range(DIM):
for j in range(DIM):
Lappsi += gammaUU[i][j] * psi_dcovDD[i][j]
# Step 9: Define right-hand sides for the evolution.
global Gamma_rhs
ArhsD = ixp.zerorank1()
ErhsD = ixp.zerorank1()
for i in range(DIM):
ArhsD[i] = -ED[i] - psi_dD[i]
ErhsD[i] = -LapAD[i] + Gamma_dD[i]
psi_rhs = -Gamma
Gamma_rhs = -Lappsi
else:
print(
"Invalid choice of system: System_to_use must be either System_I or System_II"
)
ED_dD = ixp.declarerank2("ED_dD", "nosym")
global Cviolation
Cviolation = gri.register_gridfunctions("AUX", ["Cviolation"])
Cviolation = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
Cviolation += gammaUU[i][j] * ED_dD[j][i]
for b in range(DIM):
Cviolation -= gammaUU[i][j] * GammaUDD[b][i][j] * ED[b]
def Psi4(specify_tetrad=True):
global psi4_im_pt, psi4_re_pt
# Step 1.b: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 1.c: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.d: Import all ADM quantities as written in terms of BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as AB
AB.ADM_in_terms_of_BSSN()
# Step 1.e: Set up tetrad vectors
if specify_tetrad == True:
import BSSN.Psi4_tetrads as BP4t
BP4t.Psi4_tetrads()
mre4U = BP4t.mre4U
mim4U = BP4t.mim4U
n4U = BP4t.n4U
else:
# For code validation against NRPy+ psi_4 tutorial module (Tutorial-Psi4.ipynb);
# ensures a more complete code validation.
mre4U = ixp.declarerank1("mre4U", DIM=4)
mim4U = ixp.declarerank1("mim4U", DIM=4)
n4U = ixp.declarerank1("n4U", DIM=4)
# Step 2: Construct the (rank-4) Riemann curvature tensor associated with the ADM 3-metric:
RDDDD = ixp.zerorank4()
gammaDDdDD = AB.gammaDDdDD
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
RDDDD[i][k][l][m] = sp.Rational(1, 2) * \
(gammaDDdDD[i][m][k][l] + gammaDDdDD[k][l][i][m] - gammaDDdDD[i][l][k][m] -
gammaDDdDD[k][m][i][l])
# ... then we add the term on the right:
gammaDD = AB.gammaDD
GammaUDD = AB.GammaUDD
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
for n in range(DIM):
for p in range(DIM):
RDDDD[i][k][l][m] += gammaDD[n][p] * \
(GammaUDD[n][k][l] * GammaUDD[p][i][m] - GammaUDD[n][k][m] * GammaUDD[p][i][l])
# Step 3: Construct the (rank-4) tensor in term 1 of psi_4 (referring to Eq 5.1 in
# Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf
rank4term1DDDD = ixp.zerorank4()
KDD = AB.KDD
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
rank4term1DDDD[i][j][k][l] = RDDDD[i][j][k][
l] + KDD[i][k] * KDD[l][j] - KDD[i][l] * KDD[k][j]
# Step 4: Construct the (rank-3) tensor in term 2 of psi_4 (referring to Eq 5.1 in
# Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf
rank3term2DDD = ixp.zerorank3()
KDDdD = AB.KDDdD
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
rank3term2DDD[j][k][l] = sp.Rational(
1, 2) * (KDDdD[j][k][l] - KDDdD[j][l][k])
# ... then we construct the second term in this sum:
# \Gamma^{p}_{j[k} K_{l]p} = \frac{1}{2} (\Gamma^{p}_{jk} K_{lp}-\Gamma^{p}_{jl} K_{kp}):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
for p in range(DIM):
rank3term2DDD[j][k][l] += sp.Rational(
1, 2) * (GammaUDD[p][j][k] * KDD[l][p] -
GammaUDD[p][j][l] * KDD[k][p])
# Finally, we multiply the term by $-8$:
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
rank3term2DDD[j][k][l] *= sp.sympify(-8)
# Step 5: Construct the (rank-2) tensor in term 3 of psi_4 (referring to Eq 5.1 in
# Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf
# Step 5.1: Construct 3-Ricci tensor R_{ij} = gamma^{im} R_{ijml}
RDD = ixp.zerorank2()
gammaUU = AB.gammaUU
for j in range(DIM):
for l in range(DIM):
for i in range(DIM):
for m in range(DIM):
RDD[j][l] += gammaUU[i][m] * RDDDD[i][j][m][l]
# Step 5.2: Construct K^p_l = gamma^{pi} K_{il}
KUD = ixp.zerorank2()
for p in range(DIM):
for l in range(DIM):
for i in range(DIM):
KUD[p][l] += gammaUU[p][i] * KDD[i][l]
# Step 5.3: Construct trK = gamma^{ij} K_{ij}
trK = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
trK += gammaUU[i][j] * KDD[i][j]
# Next we put these terms together to construct the entire term in parentheses:
# +4 \left(R_{jl} - K_{jp} K^p_l + K K_{jl} \right),
rank2term3DD = ixp.zerorank2()
for j in range(DIM):
for l in range(DIM):
rank2term3DD[j][l] = RDD[j][l] + trK * KDD[j][l]
for p in range(DIM):
rank2term3DD[j][l] += -KDD[j][p] * KUD[p][l]
# Finally we multiply by +4:
for j in range(DIM):
for l in range(DIM):
rank2term3DD[j][l] *= sp.sympify(4)
# Step 6: Construct real & imaginary parts of psi_4
# by contracting constituent rank 2, 3, and 4
# tensors with input tetrads mre4U, mim4U, & n4U.
def tetrad_product__Real_psi4(n, Mre, Mim, mu, nu, eta, delta):
return +n[mu] * Mre[nu] * n[eta] * Mre[delta] - n[mu] * Mim[nu] * n[
eta] * Mim[delta]
def tetrad_product__Imag_psi4(n, Mre, Mim, mu, nu, eta, delta):
return -n[mu] * Mre[nu] * n[eta] * Mim[delta] - n[mu] * Mim[nu] * n[
eta] * Mre[delta]
# We split psi_4 into three pieces, to expedite & possibly parallelize C code generation.
psi4_re_pt = [sp.sympify(0), sp.sympify(0), sp.sympify(0)]
psi4_im_pt = [sp.sympify(0), sp.sympify(0), sp.sympify(0)]
# First term:
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
psi4_re_pt[0] += rank4term1DDDD[i][j][
k][l] * tetrad_product__Real_psi4(
n4U, mre4U, mim4U, i + 1, j + 1, k + 1, l + 1)
psi4_im_pt[0] += rank4term1DDDD[i][j][
k][l] * tetrad_product__Imag_psi4(
n4U, mre4U, mim4U, i + 1, j + 1, k + 1, l + 1)
# Second term:
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
psi4_re_pt[1] += rank3term2DDD[j][k][l] * \
sp.Rational(1, 2) * (+tetrad_product__Real_psi4(n4U, mre4U, mim4U, 0, j + 1, k + 1, l + 1)
- tetrad_product__Real_psi4(n4U, mre4U, mim4U, j + 1, 0, k + 1, l + 1))
psi4_im_pt[1] += rank3term2DDD[j][k][l] * \
sp.Rational(1, 2) * (+tetrad_product__Imag_psi4(n4U, mre4U, mim4U, 0, j + 1, k + 1, l + 1)
- tetrad_product__Imag_psi4(n4U, mre4U, mim4U, j + 1, 0, k + 1, l + 1))
# Third term:
for j in range(DIM):
for l in range(DIM):
psi4_re_pt[2] += rank2term3DD[j][l] * \
(sp.Rational(1, 4) * (+tetrad_product__Real_psi4(n4U, mre4U, mim4U, 0, j + 1, 0, l + 1)
- tetrad_product__Real_psi4(n4U, mre4U, mim4U, j + 1, 0, 0, l + 1)
- tetrad_product__Real_psi4(n4U, mre4U, mim4U, 0, j + 1, l + 1, 0)
+ tetrad_product__Real_psi4(n4U, mre4U, mim4U, j + 1, 0, l + 1, 0)))
psi4_im_pt[2] += rank2term3DD[j][l] * \
(sp.Rational(1, 4) * (+tetrad_product__Imag_psi4(n4U, mre4U, mim4U, 0, j + 1, 0, l + 1)
- tetrad_product__Imag_psi4(n4U, mre4U, mim4U, j + 1, 0, 0, l + 1)
- tetrad_product__Imag_psi4(n4U, mre4U, mim4U, 0, j + 1, l + 1, 0)
+ tetrad_product__Imag_psi4(n4U, mre4U, mim4U, j + 1, 0, l + 1, 0)))
def GiRaFFE_Higher_Order():
#Step 1.0: Set the spatial dimension parameter to 3.
par.set_parval_from_str("grid::DIM", 3)
DIM = par.parval_from_str("grid::DIM")
# Step 1.1: Set the finite differencing order to 4.
#par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", 4)
thismodule = "GiRaFFE_NRPy"
# M_PI will allow the C code to substitute the correct value
M_PI = par.Cparameters("#define", thismodule, "M_PI", "")
# ADMBase defines the 4-metric in terms of the 3+1 spacetime metric quantities gamma_{ij}, beta^i, and alpha
gammaDD = ixp.register_gridfunctions_for_single_rank2("AUX",
"gammaDD",
"sym01",
DIM=3)
betaU = ixp.register_gridfunctions_for_single_rank1("AUX", "betaU", DIM=3)
alpha = gri.register_gridfunctions("AUX", "alpha")
# GiRaFFE uses the Valencia 3-velocity and A_i, which are defined in the initial data module(GiRaFFEfood)
ValenciavU = ixp.register_gridfunctions_for_single_rank1("AUX",
"ValenciavU",
DIM=3)
AD = ixp.register_gridfunctions_for_single_rank1("EVOL", "AD", DIM=3)
# B^i must be computed at each timestep within GiRaFFE so that the Valencia 3-velocity can be evaluated
BU = ixp.register_gridfunctions_for_single_rank1("AUX", "BU", DIM=3)
# <a id='step3'></a>
#
# ## Step 1.2: Build the four metric $g_{\mu\nu}$, its inverse $g^{\mu\nu}$ and spatial derivatives $g_{\mu\nu,i}$ from ADM 3+1 quantities $\gamma_{ij}$, $\beta^i$, and $\alpha$
#
# $$\label{step3}$$
# \[Back to [top](#top)\]
#
# Notice that the time evolution equation for $\tilde{S}_i$
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}
# $$
# contains $\partial_i g_{\mu \nu} = g_{\mu\nu,i}$. We will now focus on evaluating this term.
#
# The four-metric $g_{\mu\nu}$ is related to the three-metric $\gamma_{ij}$, index-lowered shift $\beta_i$, and lapse $\alpha$ by
# $$
# g_{\mu\nu} = \begin{pmatrix}
# -\alpha^2 + \beta^k \beta_k & \beta_j \\
# \beta_i & \gamma_{ij}
# \end{pmatrix}.
# $$
# This tensor and its inverse have already been built by the u0_smallb_Poynting__Cartesian.py module ([documented here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)), so we can simply load the module and import the variables.
# Step 1.2: import u0_smallb_Poynting__Cartesian.py to set
# the four metric and its inverse. This module also sets b^2 and u^0.
import u0_smallb_Poynting__Cartesian.u0_smallb_Poynting__Cartesian as u0b
u0b.compute_u0_smallb_Poynting__Cartesian(gammaDD, betaU, alpha,
ValenciavU, BU)
betaD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
betaD[i] += gammaDD[i][j] * betaU[j]
# We will now pull in the four metric and its inverse.
import BSSN.ADMBSSN_tofrom_4metric as AB4m # NRPy+: ADM/BSSN <-> 4-metric conversions
AB4m.g4DD_ito_BSSN_or_ADM("ADM")
g4DD = AB4m.g4DD
AB4m.g4UU_ito_BSSN_or_ADM("ADM")
g4UU = AB4m.g4UU
# Next we compute spatial derivatives of the metric, $\partial_i g_{\mu\nu} = g_{\mu\nu,i}$, written in terms of the three-metric, shift, and lapse. Simply taking the derivative of the expression for $g_{\mu\nu}$ above, we find
# $$
# g_{\mu\nu,l} = \begin{pmatrix}
# -2\alpha \alpha_{,l} + \beta^k_{\ ,l} \beta_k + \beta^k \beta_{k,l} & \beta_{i,l} \\
# \beta_{j,l} & \gamma_{ij,l}
# \end{pmatrix}.
# $$
#
# Notice the derivatives of the shift vector with its indexed lowered, $\beta_{i,j} = \partial_j \beta_i$. This can be easily computed in terms of the given ADMBase quantities $\beta^i$ and $\gamma_{ij}$ via:
# \begin{align}
# \beta_{i,j} &= \partial_j \beta_i \\
# &= \partial_j (\gamma_{ik} \beta^k) \\
# &= \gamma_{ik} \partial_j\beta^k + \beta^k \partial_j \gamma_{ik} \\
# \beta_{i,j} &= \gamma_{ik} \beta^k_{\ ,j} + \beta^k \gamma_{ik,j}.
# \end{align}
#
# Since this expression mixes Greek and Latin indices, we will need to store the expressions for each of the three spatial derivatives as separate variables.
#
# So, we will first set
# $$ g_{00,l} = \underbrace{-2\alpha \alpha_{,l}}_{\rm Term\ 1} + \underbrace{\beta^k_{\ ,l} \beta_k}_{\rm Term\ 2} + \underbrace{\beta^k \beta_{k,l}}_{\rm Term\ 3} $$
# Step 1.2, cont'd: Build spatial derivatives of the four metric
# Step 1.2.a: Declare derivatives of grid functions. These will be handled by FD_outputC
alpha_dD = ixp.declarerank1("alpha_dD")
betaU_dD = ixp.declarerank2("betaU_dD", "nosym")
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01")
# Step 1.2.b: These derivatives will be constructed analytically.
betaDdD = ixp.zerorank2()
g4DDdD = ixp.zerorank3(DIM=4)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
# \gamma_{ik} \beta^k_{,j} + \beta^k \gamma_{ik,j}
betaDdD[i][j] += gammaDD[i][k] * betaU_dD[k][j] + betaU[
k] * gammaDD_dD[i][k][j]
# Step 1.2.c: Set the 00 components
# Step 1.2.c.i: Term 1: -2\alpha \alpha_{,l}
for l in range(DIM):
g4DDdD[0][0][l + 1] = -2 * alpha * alpha_dD[l]
# Step 1.2.c.ii: Term 2: \beta^k_{\ ,l} \beta_k
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l + 1] += betaU_dD[k][l] * betaD[k]
# Step 1.2.c.iii: Term 3: \beta^k \beta_{k,l}
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l + 1] += betaU[k] * betaDdD[k][l]
# Now we will contruct the other components of $g_{\mu\nu,l}$. We will first construct
# $$ g_{i0,l} = g_{0i,l} = \beta_{i,l}, $$
# then
# $$ g_{ij,l} = \gamma_{ij,l} $$
# Step 1.2.d: Set the i0 and 0j components
for l in range(DIM):
for i in range(DIM):
# \beta_{i,l}
g4DDdD[i + 1][0][l + 1] = g4DDdD[0][i + 1][l + 1] = betaDdD[i][l]
#Step 1.2.e: Set the ij components
for l in range(DIM):
for i in range(DIM):
for j in range(DIM):
# \gamma_{ij,l}
g4DDdD[i + 1][j + 1][l + 1] = gammaDD_dD[i][j][l]
# <a id='step4'></a>
#
# # $T^{\mu\nu}_{\rm EM}$ and its derivatives
#
# Now that the metric and its derivatives are out of the way, let's return to the evolution equation for $\tilde{S}_i$,
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}.
# $$
# We turn our focus now to $T^j_{{\rm EM} i}$ and its derivatives. To this end, we start by computing $T^{\mu \nu}_{\rm EM}$ (from eq. 27 of [Paschalidis & Shapiro's paper on their GRFFE code](https://arxiv.org/pdf/1310.3274.pdf)):
#
# $$\boxed{T^{\mu \nu}_{\rm EM} = b^2 u^{\mu} u^{\nu} + \frac{b^2}{2} g^{\mu \nu} - b^{\mu} b^{\nu}.}$$
#
# Notice that $T^{\mu\nu}_{\rm EM}$ is written in terms of
#
# * $b^\mu$, the 4-component magnetic field vector, related to the comoving magnetic field vector $B^i_{(u)}$
# * $u^\mu$, the 4-velocity
# * $g^{\mu \nu}$, the inverse 4-metric
#
# However, $\texttt{GiRaFFE}$ has access to only the following quantities, requiring in the following sections that we write the above quantities in terms of the following ones:
#
# * $\gamma_{ij}$, the 3-metric
# * $\alpha$, the lapse
# * $\beta^i$, the shift
# * $A_i$, the vector potential
# * $B^i$, the magnetic field (we assume only in the grid interior, not the ghost zones)
# * $\left[\sqrt{\gamma}\Phi\right]$, the zero-component of the vector potential $A_\mu$, times the square root of the determinant of the 3-metric
# * $v_{(n)}^i$, the Valencia 3-velocity
# * $u^0$, the zero-component of the 4-velocity
#
# ## Step 2.0: $u^i$ and $b^i$ and related quantities
# $$\label{step4}$$
# \[Back to [top](#top)\]
#
# We begin by importing what we can from u0_smallb_Poynting__Cartesian.py. We will need the four-velocity $u^\mu$, which is related to the Valencia 3-velocity $v^i_{(n)}$ used directly by $\texttt{GiRaFFE}$ (see also [Duez, et al, eqs. 53 and 56](https://arxiv.org/pdf/astro-ph/0503420.pdf))
# \begin{align}
# u^i &= u^0 (\alpha v^i_{(n)} - \beta^i), \\
# u_j &= \alpha u^0 \gamma_{ij} v^i_{(n)},
# \end{align}
# and $v^i_{(n)}$ is the Valencia three-velocity. These have already been constructed in terms of the Valencia 3-velocity and other 3+1 ADM quantities by the u0_smallb_Poynting__Cartesian.py module, so we can simply import these variables:
# Step 2.0: u^i, b^i, and related quantities
# Step 2.0.a: import the four-velocity, as written in terms of the Valencia 3-velocity
global uD, uU
uD = ixp.register_gridfunctions_for_single_rank1("AUX", "uD")
uU = ixp.register_gridfunctions_for_single_rank1("AUX", "uU")
u4upperZero = gri.register_gridfunctions("AUX", "u4upperZero")
for i in range(DIM):
uD[i] = u0b.uD[i].subs(u0b.u0, u4upperZero)
uU[i] = u0b.uU[i].subs(u0b.u0, u4upperZero)
# We also need the magnetic field 4-vector $b^{\mu}$, which is related to the magnetic field by [eqs. 23, 24, and 31 in Duez, et al](https://arxiv.org/pdf/astro-ph/0503420.pdf):
# \begin{align}
# b^0 &= \frac{1}{\sqrt{4\pi}} B^0_{\rm (u)} = \frac{u_j B^j}{\sqrt{4\pi}\alpha}, \\
# b^i &= \frac{1}{\sqrt{4\pi}} B^i_{\rm (u)} = \frac{B^i + (u_j B^j) u^i}{\sqrt{4\pi}\alpha u^0}, \\
# \end{align}
# where $B^i$ is the variable tracked by the HydroBase thorn in the Einstein Toolkit. Again, these have already been built by the u0_smallb_Poynting__Cartesian.py module, so we can simply import the variables.
# Step 2.0.b: import the small b terms
smallb4U = ixp.zerorank1(DIM=4)
smallb4D = ixp.zerorank1(DIM=4)
for mu in range(4):
smallb4U[mu] = u0b.smallb4U[mu].subs(u0b.u0, u4upperZero)
smallb4D[mu] = u0b.smallb4D[mu].subs(u0b.u0, u4upperZero)
smallb2 = u0b.smallb2etk.subs(u0b.u0, u4upperZero)
# <a id='step5'></a>
#
# ## Step 2.1: Construct all components of the electromagnetic stress-energy tensor $T^{\mu \nu}_{\rm EM}$
# $$\label{step5}$$
#
# \[Back to [top](#top)\]
#
# We now have all the pieces to calculate the stress-energy tensor,
# $$T^{\mu \nu}_{\rm EM} = \underbrace{b^2 u^{\mu} u^{\nu}}_{\rm Term\ 1} +
# \underbrace{\frac{b^2}{2} g^{\mu \nu}}_{\rm Term\ 2}
# - \underbrace{b^{\mu} b^{\nu}}_{\rm Term\ 3}.$$
# Because $u^0$ is a separate variable, we could build the $00$ component separately, then the $\mu0$ and $0\nu$ components, and finally the $\mu\nu$ components. Alternatively, for clarity, we could create a temporary variable $u^\mu=\left( u^0, u^i \right)$
# Step 2.1: Construct the electromagnetic stress-energy tensor
# Step 2.1.a: Set up the four-velocity vector
u4U = ixp.zerorank1(DIM=4)
u4U[0] = u4upperZero
for i in range(DIM):
u4U[i + 1] = uU[i]
# Step 2.1.b: Build T4EMUU itself
T4EMUU = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
# Term 1: b^2 u^{\mu} u^{\nu}
T4EMUU[mu][nu] = smallb2 * u4U[mu] * u4U[nu]
for mu in range(4):
for nu in range(4):
# Term 2: b^2 / 2 g^{\mu \nu}
T4EMUU[mu][nu] += smallb2 * g4UU[mu][nu] / 2
for mu in range(4):
for nu in range(4):
# Term 3: -b^{\mu} b^{\nu}
T4EMUU[mu][nu] += -smallb4U[mu] * smallb4U[nu]
# <a id='step6'></a>
#
# # Step 2.2: Derivatives of the electromagnetic stress-energy tensor
# $$\label{step6}$$
#
# \[Back to [top](#top)\]
#
# If we look at the evolution equation, we see that we will need spatial derivatives of $T^{\mu\nu}_{\rm EM}$. When confronted with derivatives of complicated expressions, it is generally convenient to declare those expressions as gridfunctions themselves, allowing NRPy+ to take finite-difference derivatives of the expressions. This can even reduce the truncation error associated with the finite differences, because the alternative is to use a function of several finite-difference derivatives, allowing more error to accumulate than the extra gridfunction will introduce. While we will use that technique for some of the subexpressions of $T^{\mu\nu}_{\rm EM}$, we don't want to rely on it for the whole expression; doing so would require us to take the derivative of the magnetic field $B^i$, which is itself found by finite-differencing the vector potential $A_i$. Thus $B^i$ cannot be *consistently* defined in ghost zones. To potentially reduce numerical errors induced by inconsistent finite differencing, we will differentiate $T^{\mu\nu}_{\rm EM}$ term-by-term so that finite-difference derivatives of $A_i$ appear.
#
# We will now now take these spatial derivatives of $T^{\mu\nu}_{\rm EM}$, applying the chain rule until it is only in terms of basic gridfunctions and their derivatives: $\alpha$, $\beta^i$, $\gamma_{ij}$, $A_i$, and the four-velocity $u^i$. Along the way, we will also set up useful temporary variables representing the steps of the chain rule. (Notably, *all* of these quantities will be written in terms of $A_i$ and its derivatives):
#
# * $B^i$ (already computed in terms of $A_k$, via $B^i = \epsilon^{ijk} \partial_j A_k$),
# * $B^i_{,l}$,
# * $b^i$ and $b_i$ (already computed),
# * $b^i_{,k}$,
# * $b^2$ (already computed),
# * and $\left(b^2\right)_{,j}$.
#
# (The variables not already computed will not be seen by the ETK, as they are written in terms of $A_i$ and its derivatives; they simply help to organize the NRPy+ code.)
#
# So then,
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} &= \partial_j (g_{\mu i} T^{\mu j}_{\rm EM}) \\
# &= \partial_j \left[g_{\mu i} \left(b^2 u^j u^\mu + \frac{b^2}{2} g^{j\mu} - b^j b^\mu\right)\right] \\
# &= \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ A} + g_{\mu i} \left( \underbrace{\partial_j \left(b^2 u^j u^\mu \right)}_{\rm Term\ B} + \underbrace{\partial_j \left(\frac{b^2}{2} g^{j\mu}\right)}_{\rm Term\ C} - \underbrace{\partial_j \left(b^j b^k\right)}_{\rm Term\ D} \right) \\
# \end{align}
# Following the product and chain rules for each term, we find that
# \begin{align}
# {\rm Term\ B} &= \partial_j (b^2 u^j u^\mu) \\
# &= \partial_j b^2 u^j u^\mu + b^2 \partial_j u^j u^\mu + b^2 u^j \partial_j u^\mu \\
# &= \underbrace{\left(b^2\right)_{,j} u^j u^\mu + b^2 u^j_{,j} u^\mu + b^2 u^j u^{\mu}_{,j}}_{\rm To\ Term\ 2\ below} \\
# {\rm Term\ C} &= \partial_j \left(\frac{b^2}{2} g^{j\mu}\right) \\
# &= \frac{1}{2} \left( \partial_j b^2 g^{j\mu} + b^2 \partial_j g^{j\mu} \right) \\
# &= \underbrace{\frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} + \frac{b^2}{2} g^{j\mu}_{\ ,j}}_{\rm To\ Term\ 3\ below} \\
# {\rm Term\ D} &= \partial_j (b^j b^\mu) \\
# &= \underbrace{b^j_{,j} b^\mu + b^j b^\mu_{,j}}_{\rm To\ Term\ 2\ below}\\
# \end{align}
#
# So,
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} &= g_{\mu i,j} T^{\mu j}_{\rm EM} \\
# &+ g_{\mu i} \left(\left(b^2\right)_{,j} u^j u^\mu +b^2 u^j_{,j} u^\mu + b^2 u^j u^{\mu}_{,j} + \frac{1}{2}\left(b^2\right)_{,j} g^{j\mu} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j}\right);
# \end{align}
# We will rearrange this once more, collecting the $b^2$ terms together, noting that Term A will become Term 1:
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} =& \
# \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ 1} \\
# & + \underbrace{g_{\mu i} \left( b^2 u^j_{,j} u^\mu + b^2 u^j u^\mu_{,j} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j} \right)}_{\rm Term\ 2} \\
# & + \underbrace{g_{\mu i} \left( \left(b^2\right)_{,j} u^j u^\mu + \frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} \right).}_{\rm Term\ 3} \\
# \end{align}
#
# <a id='table2'></a>
#
# **List of Derivatives**
# $$\label{table2}$$
#
# Note that this is in terms of the derivatives of several other quantities:
#
# * [Step 2.2.a](#capitalBideriv): $B^i_{,l}$: Since $b^i$ is itself a function of $B^i$, we will first need the derivatives $B^i_{,l}$ in terms of the evolved quantity $A_i$ (the vector potential).
# * [Step 2.2.b](#bideriv): $b^i_{,k}$: Once we have $B^i_{,l}$ we can evaluate derivatives of $b^i$, $b^i_{,k}$
# * [Step 2.2.c](#b2deriv): The derivative of $b^2 = g_{\mu\nu} b^\mu b^\nu$, $\left(b^2\right)_{,j}$
# * [Step 2.2.d](#gupijderiv): Derivatives of $g^{\mu\nu}$, $g^{\mu\nu}_{\ ,k}$
# * [Step 2.2.e](#alltogether): Putting it together: $\partial_j T^{j}_{{\rm EM} i}$
# * [Step 2.2.e.i](#alltogether1): Putting it together: Term 1
# * [Step 2.2.e.ii](#alltogether2): Putting it together: Term 2
# * [Step 2.2.e.iii](#alltogether3): Putting it together: Term 3
# <a id='capitalBideriv'></a>
#
# ## Step 2.2.a: Derivatives of $B^i$
#
# $$\label{capitalbideriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# First, we will build the derivatives of the magnetic field. Since $b^i$ is a function of $B^i$, we will start from the definition of $B^i$ in terms of $A_i$, $B^i = \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k$. We will first apply the product rule, noting that the symbol $[ijk]$ consists purely of the integers $-1, 0, 1$ and thus can be treated as a constant in this process.
# \begin{align}
# B^i_{,l} &= \partial_l \left( \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k \right) \\
# &= [ijk] \partial_l \left( \frac{1}{\sqrt{\gamma}}\right) \partial_j A_k + \frac{[ijk]}{\sqrt{\gamma}} \partial_l \partial_j A_k \\
# &= [ijk]\left(-\frac{\gamma_{,l}}{2\gamma^{3/2}}\right) \partial_j A_k + \frac{[ijk]}{\sqrt{\gamma}} \partial_l \partial_j A_k \\
# \end{align}
# Now, we will substitute back in for the definition of the Levi-Civita tensor: $\epsilon^{ijk} = [ijk] / \sqrt{\gamma}$. Then we will substitute the magnetic field $B^i$ back in.
# \begin{align}
# B^i_{,l} &= -\frac{\gamma_{,l}}{2\gamma} \epsilon^{ijk} \partial_j A_k + \epsilon^{ijk} \partial_l \partial_j A_k \\
# &= -\frac{\gamma_{,l}}{2\gamma} B^i + \epsilon^{ijk} A_{k,jl}, \\
# \end{align}
#
# Thus, the expression we are left with for the derivatives of the magnetic field is:
# \begin{align}
# B^i_{,l} &= \underbrace{-\frac{\gamma_{,l}}{2\gamma} B^i}_{\rm Term\ 1} + \underbrace{\epsilon^{ijk} A_{k,jl}}_{\rm Term\ 2}, \\
# \end{align}
# where $\epsilon^{ijk} = [ijk] / \sqrt{\gamma}$ is the antisymmetric Levi-Civita tensor and $\gamma$ is the determinant of the three-metric.
#
# Step 2.2: Derivatives of the electromagnetic stress-energy tensor
# We already have a handy function to define the Levi-Civita symbol in WeylScalars
import WeylScal4NRPy.WeylScalars_Cartesian as weyl
# Initialize the Levi-Civita tensor by setting it equal to the Levi-Civita symbol
LeviCivitaSymbolDDD = weyl.define_LeviCivitaSymbol_rank3()
LeviCivitaTensorDDD = ixp.zerorank3()
LeviCivitaTensorUUU = ixp.zerorank3()
global gammaUU, gammadet
gammaUU = ixp.register_gridfunctions_for_single_rank2(
"AUX", "gammaUU", "sym01")
gammadet = gri.register_gridfunctions("AUX", "gammadet")
gammaUU, gammadet = ixp.symm_matrix_inverter3x3(gammaDD)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LeviCivitaTensorDDD[i][j][
k] = LeviCivitaSymbolDDD[i][j][k] * sp.sqrt(gammadet)
LeviCivitaTensorUUU[i][j][
k] = LeviCivitaSymbolDDD[i][j][k] / sp.sqrt(gammadet)
AD_dD = ixp.declarerank2("AD_dD", "nosym")
# Step 2.2.a: Construct the derivatives of the magnetic field.
gammadet_dD = ixp.declarerank1("gammadet_dD")
AD_dDD = ixp.declarerank3("AD_dDD", "sym12")
# The other partial derivatives of B^i
BUdD = ixp.zerorank2()
for i in range(DIM):
for l in range(DIM):
# Term 1: -\gamma_{,l} / (2\gamma) B^i
BUdD[i][l] = -gammadet_dD[l] * BU[i] / (2 * gammadet)
for i in range(DIM):
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
# Term 2: \epsilon^{ijk} A_{k,jl}
BUdD[i][
l] += LeviCivitaTensorUUU[i][j][k] * AD_dDD[k][j][l]
# <a id='bideriv'></a>
#
# ## Step 2.2.b: Derivatives of $b^i$
# $$\label{bideriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Now, we will code the derivatives of the spatial components of $b^{\mu}$, $b^i$:
# $$
# b^i_{,k} = \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \left(B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2}.
# $$
#
# We should note that while $b^\mu$ is a four-vector (and the code reflects this: $\text{smallb4U}$ and $\text{smallb4U}$ have $\text{DIM=4}$), we only need the spatial components. We will only focus on the spatial components for the moment.
#
#
# Let's go into a little more detail on where this comes from. We start from the definition $$b^i = \frac{B^i + (u_j B^j) u^i}{\sqrt{4\pi}\alpha u^0};$$ we then apply the quotient rule:
# \begin{align}
# b^i_{,k} &= \frac{\left(\sqrt{4\pi}\alpha u^0\right) \partial_k \left(B^i + (u_j B^j) u^i\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\sqrt{4\pi}\alpha u^0\right)}{\left(\sqrt{4\pi}\alpha u^0\right)^2} \\
# &= \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \partial_k \left(B^i + (u_j B^j) u^i\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2} \\
# \end{align}
# Note that $\left( \alpha u^0 \right)$ is being used as its own gridfunction, so $\partial_k \left(a u^0\right)$ will be finite-differenced by NRPy+ directly. We will also apply the product rule to the term $\partial_k \left(B^i + (u_j B^j) u^i\right) = B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}$. So,
# $$ b^i_{,k} = \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \left(B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2}. $$
#
# It will be easier to code this up if we rearrange these terms to group together the terms that involve contractions over $j$. Doing that, we find
# $$
# b^i_{,k} = \frac{\overbrace{\alpha u^0 B^i_{,k} - B^i \partial_k (\alpha u^0)}^{\rm Term\ Num1} + \overbrace{\left( \alpha u^0 \right) \left( u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k} \right)}^{\rm Term\ Num2.a} - \overbrace{\left( u_j B^j u^i \right) \partial_k \left( \alpha u^0 \right) }^{\rm Term\ Num2.b}}{\underbrace{\sqrt{4 \pi} \left( \alpha u^0 \right)^2}_{\rm Term\ Denom}}.
# $$
global u0alpha
u0alpha = gri.register_gridfunctions("AUX", "u0alpha")
u0alpha = alpha * u4upperZero
u0alpha_dD = ixp.declarerank1("u0alpha_dD")
uU_dD = ixp.declarerank2("uU_dD", "nosym")
uD_dD = ixp.declarerank2("uD_dD", "nosym")
# Step 2.2.b: Construct derivatives of the small b vector
# smallbUdD represents the derivative of smallb4U
smallbUdD = ixp.zerorank2()
for i in range(DIM):
for k in range(DIM):
# Term Num1: \alpha u^0 B^i_{,k} - B^i \partial_k (\alpha u^0)
smallbUdD[i][k] += u0alpha * BUdD[i][k] - BU[i] * u0alpha_dD[k]
for i in range(DIM):
for k in range(DIM):
for j in range(DIM):
# Term Num2.a: terms that require contractions over k, and thus an extra loop.
# ( \alpha u^0 ) ( u_{j,k} B^j u^i
# + u_j B^j_{,k} u^i
# + u_j B^j u^i_{,k} )
smallbUdD[i][k] += u0alpha * (uD_dD[j][k] * BU[j] * uU[i] +
uD[j] * BUdD[j][k] * uU[i] +
uD[j] * BU[j] * uU_dD[i][k])
for i in range(DIM):
for k in range(DIM):
for j in range(DIM):
#Term 2.b (More contractions over k): ( u_j B^j u^i ) ( \alpha u^0 ),k
smallbUdD[i][k] += -(uD[j] * BU[j] * uU[i]) * u0alpha_dD[k]
for i in range(DIM):
for k in range(DIM):
# Term Denom: Divide the numerator by sqrt(4 pi) * (alpha u^0)^2
smallbUdD[i][k] /= sp.sqrt(4 * M_PI) * u0alpha * u0alpha
# <a id='b2deriv'></a>
#
# ## Step 2.2.c: Derivative of $b^2$
# $$\label{b2deriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Here, we will take the derivative of $b^2 = g_{\mu\nu} b^\mu b^\nu$. Using the product rule,
# \begin{align}
# \left(b^2\right)_{,j} &= \partial_j \left( g_{\mu\nu} b^\mu b^\nu \right) \\
# &= g_{\mu\nu,j} b^\mu b^\nu + g_{\mu\nu} b^\mu_{,j} b^\nu + g_{\mu\nu} b^\mu b^\nu_{,j} \\
# &= g_{\mu\nu,j} b^\mu b^\nu + 2 g_{\mu\nu} b^\mu_{,j} b^\nu.
# \end{align}
# We have already defined the spatial derivatives of the four-metric $g_{\mu\nu,j}$ in [this section](#step3); we have also defined the spatial derivatives of spatial components of $b^\mu$, $b^i_{,k}$ in [this section](#bideriv). Notice the above expression requires spatial derivatives of the *zeroth* component of $b^\mu$ as well, $b^0_{,j}$, which we will now compute. Starting with the definition, and applying the quotient rule:
# \begin{align}
# b^0 &= \frac{u_k B^k}{\sqrt{4\pi}\alpha}, \\
# \rightarrow b^0_{,j} &= \frac{1}{\sqrt{4\pi}} \frac{\alpha \left( u_{k,j} B^k + u_k B^k_{,j} \right) - u_k B^k \alpha_{,j}}{\alpha^2} \\
# &= \frac{\alpha u_{k,j} B^k + \alpha u_k B^k_{,j} - \alpha_{,j} u_k B^k}{\sqrt{4\pi} \alpha^2}.
# \end{align}
# We will first code the numerator, and then divide through by the denominator.
# Step 2.2.c: Construct the derivative of b^2
# First construct the derivative b^0_{,j}
# This four-vector will make b^2 simpler:
smallb4UdD = ixp.zerorank2(DIM=4)
# Fill in the zeroth component
for j in range(DIM):
for k in range(DIM):
# The numerator: \alpha u_{k,j} B^k
# + \alpha u_k B^k_{,j}
# - \alpha_{,j} u_k B^k
smallb4UdD[0][j + 1] += alpha * uD_dD[k][j] * BU[k] + alpha * uD[
k] * BUdD[k][j] - alpha_dD[j] * uD[k] * BU[k]
for j in range(DIM):
# Divide through by the denominator: \sqrt{4\pi} \alpha^2
smallb4UdD[0][j + 1] /= sp.sqrt(4 * M_PI) * alpha * alpha
# At this point, both $b^0_{\ ,j}$ and $b^i_{\ ,j}$ have been computed, but one exists inconveniently in the $4\times 4$ component $\verb|smallb4UdD[][]|$ and the other in the $3\times 3$ component $\verb|smallbUdD[][]|$. So that we can perform full implied sums over $g_{\mu\nu} b^\mu_{,j} b^\nu$ more conveniently, we will now store all information from $\verb|smallbUdD[i][j]|$ into $\verb|smallb4UdD[i+1][j+1]|$:
# Now, we'll fill out the rest of the four-vector with b^i_{,j} that we derived above.
for i in range(DIM):
for j in range(DIM):
smallb4UdD[i + 1][j + 1] = smallbUdD[i][j]
# Using 4-component (Greek-indexed) quantities, we can now complete our construction of
# $$\left(b^2\right)_{,j} = g_{\mu\nu,j} b^\mu b^\nu + 2 g_{\mu\nu} b^\mu_{,j} b^\nu:$$
smallb2_dD = ixp.zerorank1()
for j in range(DIM):
for mu in range(4):
for nu in range(4):
# g_{\mu\nu,j} b^\mu b^\nu
# + 2 g_{\mu\nu} b^\mu_{,j} b^\nu
smallb2_dD[j] += g4DDdD[mu][nu][j + 1] * smallb4U[
mu] * smallb4U[nu] + 2 * g4DD[mu][nu] * smallb4UdD[mu][
j + 1] * smallb4U[nu]
# <a id='gupijderiv'></a>
#
# ## Step 2.2.d: Derivatives of $g^{\mu\nu}$
# $$\label{gupijderiv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will need derivatives of the inverse four-metric, as well. Let us begin with $g^{00}$: since $g^{00} = -1/\alpha^2$ ([Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf)), $$g^{00}_{\ ,k} = \frac{2 \alpha_{,k}}{\alpha^3}$$
#
# Step 2.2.d: Construct derivatives of the components of g^{\mu\nu}
g4UUdD = ixp.zerorank3(DIM=4)
for k in range(DIM):
# 2 \alpha_{,k} / \alpha^3
g4UUdD[0][0][k + 1] = 2 * alpha_dD[k] / alpha**3
# Now, we will code the $g^{i0}_{\ ,k}$ and $g^{0j}_{\ ,k}$ components. According to [Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf), $g^{i0} = g^{0i} = \beta^i/\alpha^2$, so $$g^{i0}_{\ ,k} = g^{0i}_{\ ,k} = \frac{\alpha^2 \beta^i_{,k} - 2 \beta^i \alpha \alpha_{,k}}{\alpha^4}$$ by the quotient rule. So, we'll code
# $$
# g^{i0} = g^{0i} =
# \underbrace{\frac{\beta^i_{,k}}{\alpha^2}}_{\rm Term\ 1}
# - \underbrace{\frac{2 \beta^i \alpha_{,k}}{\alpha^3}}_{\rm Term\ 2}
# $$
for k in range(DIM):
for i in range(DIM):
# Term 1: \beta^i_{,k} / \alpha^2
g4UUdD[i + 1][0][k +
1] = g4UUdD[0][i +
1][k +
1] = betaU_dD[i][k] / alpha**2
for k in range(DIM):
for i in range(DIM):
# Term 2: -2 \beta^i \alpha_{,k} / \alpha^3
g4UUdD[i + 1][0][k + 1] += -2 * betaU[i] * alpha_dD[k] / alpha**3
g4UUdD[0][i + 1][k + 1] += -2 * betaU[i] * alpha_dD[k] / alpha**3
# We will also need derivatives of the spatial part of the inverse four-metric: since $g^{ij} = \gamma^{ij} - \frac{\beta^i \beta^j}{\alpha^2}$ ([Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf)),
# \begin{align}
# g^{ij}_{\ ,k} &= \gamma^{ij}_{\ ,k} - \frac{\alpha^2 \partial_k (\beta^i \beta^j) - \beta^i \beta^j \partial_k \alpha^2}{(\alpha^2)^2} \\
# &= \gamma^{ij}_{\ ,k} - \frac{\alpha^2\beta^i \beta^j_{,k}+\alpha^2\beta^i_{,k} \beta^j-2\beta^i \beta^j \alpha \alpha_{,k}}{\alpha^4}. \\
# &= \gamma^{ij}_{\ ,k} - \frac{\alpha\beta^i \beta^j_{,k}+\alpha\beta^i_{,k} \beta^j-2\beta^i \beta^j \alpha_{,k}}{\alpha^3} \\
# g^{ij}_{\ ,k} &= \underbrace{\gamma^{ij}_{\ ,k}}_{\rm Term\ 1} - \underbrace{\frac{\beta^i \beta^j_{,k}}{\alpha^2}}_{\rm Term\ 2} - \underbrace{\frac{\beta^i_{,k} \beta^j}{\alpha^2}}_{\rm Term\ 3} + \underbrace{\frac{2\beta^i \beta^j \alpha_{,k}}{\alpha^3}}_{\rm Term\ 4}. \\
# \end{align}
#
gammaUU_dD = ixp.declarerank3("gammaUU_dD", "sym01")
# The spatial derivatives of the spatial components of the four metric:
# Term 1: \gamma^{ij}_{\ ,k}
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j + 1][k + 1] = gammaUU_dD[i][j][k]
# Term 2: - \beta^i \beta^j_{,k} / \alpha^2
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j +
1][k +
1] += -betaU[i] * betaU_dD[j][k] / alpha**2
# Term 3: - \beta^i_{,k} \beta^j / \alpha^2
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j +
1][k +
1] += -betaU_dD[i][k] * betaU[j] / alpha**2
# Term 4: 2\beta^i \beta^j \alpha_{,k}\alpha^3
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j + 1][
k + 1] += 2 * betaU[i] * betaU[j] * alpha_dD[k] / alpha**3
# <a id='alltogether'></a>
#
# ## Step 2.2.e: Putting it together:
# $$\label{alltogether}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# So, we can now put it all together, starting from the expression we derived above in [Step 2.2](#step6):
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} =& \
# \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ 1} \\
# & + \underbrace{g_{\mu i} \left( b^2 u^j_{,j} u^\mu + b^2 u^j u^\mu_{,j} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j} \right)}_{\rm Term\ 2} \\
# & + \underbrace{g_{\mu i} \left( \left(b^2\right)_{,j} u^j u^\mu + \frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} \right).}_{\rm Term\ 3} \\
# \end{align}
#
# <a id='alltogether1'></a>
#
# ### Step 2.2.e.i: Putting it together: Term 1
# $$\label{alltogether1}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will now construct this term by term. Term 1 is straightforward: $${\rm Term\ 1} = \gamma_{\mu i,j} T^{\mu j}_{\rm EM}.$$
# Step 2.2.e: Construct TEMUDdD_contracted itself
# Step 2.2.e.i
TEMUDdD_contracted = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 1: g_{\mu i,j} T^{\mu j}_{\rm EM}
TEMUDdD_contracted[i] += g4DDdD[mu][i + 1][j +
1] * T4EMUU[mu][j +
1]
# We'll need derivatives of u4U for the next part:
u4UdD = ixp.zerorank2(DIM=4)
u4upperZero_dD = ixp.declarerank1(
"u4upperZero_dD"
) # Note that derivatives can't be done in 4-D with the current version of NRPy
for i in range(DIM):
u4UdD[0][i + 1] = u4upperZero_dD[i]
for i in range(DIM):
for j in range(DIM):
u4UdD[i + 1][j + 1] = uU_dD[i][j]
# <a id='alltogether2'></a>
#
# ### Step 2.2.e.ii: Putting it together: Term 2
# $$\label{alltogether2}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will now add $${\rm Term\ 2} = g_{\mu i} \left( \underbrace{b^2 u^j_{,j} u^\mu}_{\rm Term\ 2a} + \underbrace{b^2 u^j u^\mu_{,j}}_{\rm Term\ 2b} + \underbrace{\frac{b^2}{2} g^{j\mu}_{\ ,j}}_{\rm Term\ 2c} + \underbrace{b^j_{,j} b^\mu}_{\rm Term\ 2d} + \underbrace{b^j b^\mu_{,j}}_{\rm Term\ 2e} \right)$$ to $\partial_j T^{j}_{{\rm EM} i}$. These are the terms that involve contractions over $k$ (but no metric derivatives like Term 1 had).
#
# Step 2.2.e.ii
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2a: g_{\mu i} b^2 u^j_{,j} u^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2 * uU_dD[j][j] * u4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2b: g_{\mu i} b^2 u^j u^\mu_{,j}
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2 * uU[j] * u4UdD[mu][j + 1]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2c: g_{\mu i} b^2 g^{j \mu}_{,j} / 2
TEMUDdD_contracted[i] += g4DD[mu][i + 1] * smallb2 * g4UUdD[
j + 1][mu][j + 1] / 2
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2d: g_{\mu i} b^j_{,j} b^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallbUdD[j][j] * smallb4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2e: g_{\mu i} b^j b^\mu_{,j}
TEMUDdD_contracted[i] += g4DD[mu][i + 1] * smallb4U[
j + 1] * smallb4UdD[mu][j + 1]
# <a id='alltogether3'></a>
#
# ### Step 2.2.e.iii: Putting it together: Term 3
# $$\label{alltogether3}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Now, we will add $${\rm Term\ 3} = g_{\mu i} \left( \underbrace{\left(b^2\right)_{,j} u^j u^\mu}_{\rm Term\ 3a} + \underbrace{\frac{1}{2} \left(b^2\right)_{,j} g^{j\mu}}_{\rm Term\ 3b} \right).$$
# Step 2.2.e.iii
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 3a: g_{\mu i} ( b^2 )_{,j} u^j u^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2_dD[j] * uU[j] * u4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 3b: g_{mu i} ( b^2 )_{,j} g^{j\mu} / 2
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2_dD[j] * g4UU[j + 1][mu] / 2
#
# # Evolution equation for $\tilde{S}_i$
# <a id='step7'></a>
#
# ## Step 3.0: Construct the evolution equation for $\tilde{S}_i$
# $$\label{step7}$$
#
# \[Back to [top](#top)\]
#
# Finally, we will return our attention to the time evolution equation (from eq. 13 of the [original paper](https://arxiv.org/pdf/1704.00599.pdf)),
# \begin{align}
# \partial_t \tilde{S}_i &= - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} \\
# &= -T^j_{{\rm EM} i} \partial_j (\alpha \sqrt{\gamma}) - \alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i} + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} \\
# &= \underbrace{-g_{i\mu} T^{\mu j}_{\rm EM} \partial_j (\alpha \sqrt{\gamma})
# }_{\rm Term\ 1} - \underbrace{\alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i}}_{\rm Term\ 2} + \underbrace{\frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}}_{\rm Term\ 3} .
# \end{align}
# We will first take derivatives of $\alpha \sqrt{\gamma}$, then construct each term in turn.
# Step 3.0: Construct the evolution equation for \tilde{S}_i
# Here, we set up the necessary machinery to take FD derivatives of alpha * sqrt(gamma)
global alpsqrtgam
alpsqrtgam = gri.register_gridfunctions("AUX", "alpsqrtgam")
alpsqrtgam = alpha * sp.sqrt(gammadet)
alpsqrtgam_dD = ixp.declarerank1("alpsqrtgam_dD")
global Stilde_rhsD
Stilde_rhsD = ixp.zerorank1()
# The first term: g_{i\mu} T^{\mu j}_{\rm EM} \partial_j (\alpha \sqrt{\gamma})
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
Stilde_rhsD[i] += -g4DD[i + 1][mu] * T4EMUU[mu][
j + 1] * alpsqrtgam_dD[j]
# The second term: \alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i}
for i in range(DIM):
Stilde_rhsD[i] += -alpsqrtgam * TEMUDdD_contracted[i]
# The third term: \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} / 2
for i in range(DIM):
for mu in range(4):
for nu in range(4):
Stilde_rhsD[i] += alpsqrtgam * T4EMUU[mu][nu] * g4DDdD[mu][nu][
i + 1] / 2
# # Evolution equations for $A_i$ and $\Phi$
# <a id='step8'></a>
#
# ## Step 4.0: Construct the evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$
# $$\label{step8}$$
#
# \[Back to [top](#top)\]
#
# We will also need to evolve the vector potential $A_i$. This evolution is given as eq. 17 in the [$\texttt{GiRaFFE}$](https://arxiv.org/pdf/1704.00599.pdf) paper:
# $$\boxed{\partial_t A_i = \epsilon_{ijk} v^j B^k - \partial_i (\underbrace{\alpha \Phi - \beta^j A_j}_{\rm AevolParen}),}$$
# where $\epsilon_{ijk} = [ijk] \sqrt{\gamma}$ is the antisymmetric Levi-Civita tensor, the drift velocity $v^i = u^i/u^0$, $\gamma$ is the determinant of the three metric, $B^k$ is the magnetic field, $\alpha$ is the lapse, and $\beta$ is the shift.
# The scalar electric potential $\Phi$ is also evolved by eq. 19:
# $$\boxed{\partial_t [\sqrt{\gamma} \Phi] = -\partial_j (\underbrace{\alpha\sqrt{\gamma}A^j - \beta^j [\sqrt{\gamma} \Phi]}_{\rm PevolParenU[j]}) - \xi \alpha [\sqrt{\gamma} \Phi],}$$
# with $\xi$ chosen as a damping factor.
#
# ### Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
#
# After declaring a some needed quantities, we will also define the parenthetical terms (underbrace above) that we need to take derivatives of. That way, we can take finite-difference derivatives easily. Note that we use $A^j = \gamma^{ij} A_i$, while $A_i$ (with $\Phi$) is technically a four-vector; this is justified, however, since $n_\mu A^\mu = 0$, where $n_\mu$ is a normal to the hypersurface, $A^0=0$ (according to Sec. II, subsection C of [this paper](https://arxiv.org/pdf/1110.4633.pdf)).
# Step 4.0: Construct the evolution equations for A_i and sqrt(gamma)Phi
# Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
xi = par.Cparameters(
"REAL", thismodule, "xi", 0.1
) # The (dimensionful) Lorenz damping factor. Recommendation: set to ~1.5/max(delta t).
# Define sqrt(gamma)Phi as psi6Phi
psi6Phi = gri.register_gridfunctions("EVOL", "psi6Phi")
Phi = psi6Phi / sp.sqrt(gammadet)
# We'll define a few extra gridfunctions to avoid complicated derivatives
global AevolParen, PevolParenU
AevolParen = gri.register_gridfunctions("AUX", "AevolParen")
PevolParenU = ixp.register_gridfunctions_for_single_rank1(
"AUX", "PevolParenU")
# {\rm AevolParen} = \alpha \Phi - \beta^j A_j
AevolParen = alpha * Phi
for j in range(DIM):
AevolParen += -betaU[j] * AD[j]
# {\rm PevolParenU[j]} = \alpha\sqrt{\gamma} \gamma^{ij} A_i - \beta^j [\sqrt{\gamma} \Phi]
for j in range(DIM):
PevolParenU[j] = -betaU[j] * psi6Phi
for i in range(DIM):
PevolParenU[j] += alpha * sp.sqrt(gammadet) * gammaUU[i][j] * AD[i]
AevolParen_dD = ixp.declarerank1("AevolParen_dD")
PevolParenU_dD = ixp.declarerank2("PevolParenU_dD", "nosym")
# ### Step 4.0.b: Construct the actual evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$
#
# Now to set the evolution equations ([eqs. 17 and 19](https://arxiv.org/pdf/1704.00599.pdf)), recalling that the drift velocity $v^i = u^i/u^0$:
# \begin{align}
# \partial_t A_i &= \epsilon_{ijk} v^j B^k - \partial_i (\alpha \Phi - \beta^j A_j) \\
# &= \epsilon_{ijk} \frac{u^j}{u^0} B^k - {\rm AevolParen\_dD[i]} \\
# \partial_t [\sqrt{\gamma} \Phi] &= -\partial_j \left(\left(\alpha\sqrt{\gamma}\right)A^j - \beta^j [\sqrt{\gamma} \Phi]\right) - \xi \alpha [\sqrt{\gamma} \Phi] \\
# &= -{\rm PevolParenU\_dD[j][j]} - \xi \alpha [\sqrt{\gamma} \Phi]. \\
# \end{align}
# Step 4.0.b: Construct the actual evolution equations for A_i and sqrt(gamma)Phi
global A_rhsD, psi6Phi_rhs
A_rhsD = ixp.zerorank1()
psi6Phi_rhs = sp.sympify(0)
for i in range(DIM):
A_rhsD[i] = -AevolParen_dD[i]
for j in range(DIM):
for k in range(DIM):
A_rhsD[i] += LeviCivitaTensorDDD[i][j][k] * (
uU[j] / u4upperZero) * BU[k]
psi6Phi_rhs = -xi * alpha * psi6Phi
for j in range(DIM):
psi6Phi_rhs += -PevolParenU_dD[j][j]
def GiRaFFE_NRPy_Main_Driver_generate_all(out_dir):
cmd.mkdir(out_dir)
gammaDD = ixp.register_gridfunctions_for_single_rank2("AUXEVOL",
"gammaDD",
"sym01",
DIM=3)
betaU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"betaU",
DIM=3)
alpha = gri.register_gridfunctions("AUXEVOL", "alpha")
AD = ixp.register_gridfunctions_for_single_rank1("EVOL", "AD")
BU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL", "BU")
ValenciavU = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "ValenciavU")
psi6Phi = gri.register_gridfunctions("EVOL", "psi6Phi")
StildeD = ixp.register_gridfunctions_for_single_rank1("EVOL", "StildeD")
ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"PhievolParenU",
DIM=3)
gri.register_gridfunctions("AUXEVOL", "AevolParen")
# Declare this symbol:
sqrt4pi = par.Cparameters("REAL", thismodule, "sqrt4pi", "sqrt(4.0*M_PI)")
GRHD.compute_sqrtgammaDET(gammaDD)
GRFFE.compute_AD_source_term_operand_for_FD(GRHD.sqrtgammaDET, betaU,
alpha, psi6Phi, AD)
GRFFE.compute_psi6Phi_rhs_flux_term_operand(gammaDD, GRHD.sqrtgammaDET,
betaU, alpha, AD, psi6Phi)
parens_to_print = [\
lhrh(lhs=gri.gfaccess("auxevol_gfs","AevolParen"),rhs=GRFFE.AevolParen),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","PhievolParenU0"),rhs=GRFFE.PhievolParenU[0]),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","PhievolParenU1"),rhs=GRFFE.PhievolParenU[1]),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","PhievolParenU2"),rhs=GRFFE.PhievolParenU[2]),\
]
subdir = "RHSs"
cmd.mkdir(os.path.join(out_dir, subdir))
desc = "Calculate quantities to be finite-differenced for the GRFFE RHSs"
name = "calculate_AD_gauge_term_psi6Phi_flux_term_for_RHSs"
outCfunction(
outfile=os.path.join(out_dir, subdir, name + ".h"),
desc=desc,
name=name,
params=
"const paramstruct *restrict params,const REAL *restrict in_gfs,REAL *restrict auxevol_gfs",
body=fin.FD_outputC("returnstring", parens_to_print,
params=outCparams).replace("IDX4", "IDX4S"),
loopopts="AllPoints",
rel_path_for_Cparams=os.path.join("../"))
xi_damping = par.Cparameters("REAL", thismodule, "xi_damping", 0.1)
GRFFE.compute_psi6Phi_rhs_damping_term(alpha, psi6Phi, xi_damping)
AevolParen_dD = ixp.declarerank1("AevolParen_dD", DIM=3)
PhievolParenU_dD = ixp.declarerank2("PhievolParenU_dD", "nosym", DIM=3)
A_rhsD = ixp.zerorank1()
psi6Phi_rhs = GRFFE.psi6Phi_damping
for i in range(3):
A_rhsD[i] += -AevolParen_dD[i]
psi6Phi_rhs += -PhievolParenU_dD[i][i]
# Add Kreiss-Oliger dissipation to the GRFFE RHSs:
# psi6Phi_dKOD = ixp.declarerank1("psi6Phi_dKOD")
# AD_dKOD = ixp.declarerank2("AD_dKOD","nosym")
# for i in range(3):
# psi6Phi_rhs += diss_strength*psi6Phi_dKOD[i]*rfm.ReU[i] # ReU[i] = 1/scalefactor_orthog_funcform[i]
# for j in range(3):
# A_rhsD[j] += diss_strength*AD_dKOD[j][i]*rfm.ReU[i] # ReU[i] = 1/scalefactor_orthog_funcform[i]
RHSs_to_print = [\
lhrh(lhs=gri.gfaccess("rhs_gfs","AD0"),rhs=A_rhsD[0]),\
lhrh(lhs=gri.gfaccess("rhs_gfs","AD1"),rhs=A_rhsD[1]),\
lhrh(lhs=gri.gfaccess("rhs_gfs","AD2"),rhs=A_rhsD[2]),\
lhrh(lhs=gri.gfaccess("rhs_gfs","psi6Phi"),rhs=psi6Phi_rhs),\
]
desc = "Calculate AD gauge term and psi6Phi RHSs"
name = "calculate_AD_gauge_psi6Phi_RHSs"
source_Ccode = outCfunction(
outfile="returnstring",
desc=desc,
name=name,
params=
"const paramstruct *params,const REAL *in_gfs,const REAL *auxevol_gfs,REAL *rhs_gfs",
body=fin.FD_outputC("returnstring", RHSs_to_print,
params=outCparams).replace("IDX4", "IDX4S"),
loopopts="InteriorPoints",
rel_path_for_Cparams=os.path.join("../")).replace(
"= NGHOSTS", "= NGHOSTS_A2B").replace(
"NGHOSTS+Nxx0", "Nxx_plus_2NGHOSTS0-NGHOSTS_A2B").replace(
"NGHOSTS+Nxx1", "Nxx_plus_2NGHOSTS1-NGHOSTS_A2B").replace(
"NGHOSTS+Nxx2", "Nxx_plus_2NGHOSTS2-NGHOSTS_A2B")
# Note the above .replace() functions. These serve to expand the loop range into the ghostzones, since
# the second-order FD needs fewer than some other algorithms we use do.
with open(os.path.join(out_dir, subdir, name + ".h"), "w") as file:
file.write(source_Ccode)
source.write_out_functions_for_StildeD_source_term(
os.path.join(out_dir, subdir), outCparams, gammaDD, betaU, alpha,
ValenciavU, BU, sqrt4pi)
subdir = "FCVAL"
cmd.mkdir(os.path.join(out_dir, subdir))
FCVAL.GiRaFFE_NRPy_FCVAL(os.path.join(out_dir, subdir))
subdir = "PPM"
cmd.mkdir(os.path.join(out_dir, subdir))
PPM.GiRaFFE_NRPy_PPM(os.path.join(out_dir, subdir))
# We will pass values of the gridfunction on the cell faces into the function. This requires us
# to declare them as C parameters in NRPy+. We will denote this with the _face infix/suffix.
alpha_face = gri.register_gridfunctions("AUXEVOL", "alpha_face")
gamma_faceDD = ixp.register_gridfunctions_for_single_rank2(
"AUXEVOL", "gamma_faceDD", "sym01")
beta_faceU = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "beta_faceU")
# We'll need some more gridfunctions, now, to represent the reconstructions of BU and ValenciavU
# on the right and left faces
Valenciav_rU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"Valenciav_rU",
DIM=3)
B_rU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"B_rU",
DIM=3)
Valenciav_lU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"Valenciav_lU",
DIM=3)
B_lU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"B_lU",
DIM=3)
subdir = "RHSs"
Af.GiRaFFE_NRPy_Afield_flux(os.path.join(out_dir, subdir))
Sf.generate_C_code_for_Stilde_flux(os.path.join(out_dir,
subdir), True, alpha_face,
gamma_faceDD, beta_faceU, Valenciav_rU,
B_rU, Valenciav_lU, B_lU, sqrt4pi)
subdir = "boundary_conditions"
cmd.mkdir(os.path.join(out_dir, subdir))
BC.GiRaFFE_NRPy_BCs(os.path.join(out_dir, subdir))
subdir = "A2B"
cmd.mkdir(os.path.join(out_dir, subdir))
A2B.GiRaFFE_NRPy_A2B(os.path.join(out_dir, subdir), gammaDD, AD, BU)
C2P_P2C.GiRaFFE_NRPy_C2P(StildeD, BU, gammaDD, betaU, alpha)
values_to_print = [\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD0"),rhs=C2P_P2C.outStildeD[0]),\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD1"),rhs=C2P_P2C.outStildeD[1]),\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD2"),rhs=C2P_P2C.outStildeD[2]),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","ValenciavU0"),rhs=C2P_P2C.ValenciavU[0]),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","ValenciavU1"),rhs=C2P_P2C.ValenciavU[1]),\
lhrh(lhs=gri.gfaccess("auxevol_gfs","ValenciavU2"),rhs=C2P_P2C.ValenciavU[2])\
]
subdir = "C2P"
cmd.mkdir(os.path.join(out_dir, subdir))
desc = "Apply fixes to \tilde{S}_i and recompute the velocity to match with current sheet prescription."
name = "GiRaFFE_NRPy_cons_to_prims"
outCfunction(
outfile=os.path.join(out_dir, subdir, name + ".h"),
desc=desc,
name=name,
params=
"const paramstruct *params,REAL *xx[3],REAL *auxevol_gfs,REAL *in_gfs",
body=fin.FD_outputC("returnstring", values_to_print,
params=outCparams).replace("IDX4", "IDX4S"),
loopopts="AllPoints,Read_xxs",
rel_path_for_Cparams=os.path.join("../"))
C2P_P2C.GiRaFFE_NRPy_P2C(gammaDD, betaU, alpha, ValenciavU, BU, sqrt4pi)
values_to_print = [\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD0"),rhs=C2P_P2C.StildeD[0]),\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD1"),rhs=C2P_P2C.StildeD[1]),\
lhrh(lhs=gri.gfaccess("in_gfs","StildeD2"),rhs=C2P_P2C.StildeD[2]),\
]
desc = "Recompute StildeD after current sheet fix to Valencia 3-velocity to ensure consistency between conservative & primitive variables."
name = "GiRaFFE_NRPy_prims_to_cons"
outCfunction(
outfile=os.path.join(out_dir, subdir, name + ".h"),
desc=desc,
name=name,
params="const paramstruct *params,REAL *auxevol_gfs,REAL *in_gfs",
body=fin.FD_outputC("returnstring", values_to_print,
params=outCparams).replace("IDX4", "IDX4S"),
loopopts="AllPoints",
rel_path_for_Cparams=os.path.join("../"))
# Write out the main driver itself:
with open(os.path.join(out_dir, "GiRaFFE_NRPy_Main_Driver.h"),
"w") as file:
file.write(r"""// Structure to track ghostzones for PPM:
typedef struct __gf_and_gz_struct__ {
REAL *gf;
int gz_lo[4],gz_hi[4];
} gf_and_gz_struct;
// Some additional constants needed for PPM:
const int VX=0,VY=1,VZ=2,BX=3,BY=4,BZ=5;
const int NUM_RECONSTRUCT_GFS = 6;
// Include ALL functions needed for evolution
#include "RHSs/calculate_AD_gauge_term_psi6Phi_flux_term_for_RHSs.h"
#include "RHSs/calculate_AD_gauge_psi6Phi_RHSs.h"
#include "PPM/reconstruct_set_of_prims_PPM_GRFFE_NRPy.c"
#include "FCVAL/interpolate_metric_gfs_to_cell_faces.h"
#include "RHSs/calculate_StildeD0_source_term.h"
#include "RHSs/calculate_StildeD1_source_term.h"
#include "RHSs/calculate_StildeD2_source_term.h"
#include "../calculate_E_field_flat_all_in_one.h"
#include "RHSs/calculate_Stilde_flux_D0.h"
#include "RHSs/calculate_Stilde_flux_D1.h"
#include "RHSs/calculate_Stilde_flux_D2.h"
#include "boundary_conditions/GiRaFFE_boundary_conditions.h"
#include "A2B/driver_AtoB.h"
#include "C2P/GiRaFFE_NRPy_cons_to_prims.h"
#include "C2P/GiRaFFE_NRPy_prims_to_cons.h"
void override_BU_with_old_GiRaFFE(const paramstruct *restrict params,REAL *restrict auxevol_gfs,const int n) {
#include "set_Cparameters.h"
char filename[100];
sprintf(filename,"BU0_override-%08d.bin",n);
FILE *out2D = fopen(filename, "rb");
fread(auxevol_gfs+BU0GF*Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,
sizeof(double),Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,out2D);
fclose(out2D);
sprintf(filename,"BU1_override-%08d.bin",n);
out2D = fopen(filename, "rb");
fread(auxevol_gfs+BU1GF*Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,
sizeof(double),Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,out2D);
fclose(out2D);
sprintf(filename,"BU2_override-%08d.bin",n);
out2D = fopen(filename, "rb");
fread(auxevol_gfs+BU2GF*Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,
sizeof(double),Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2,out2D);
fclose(out2D);
}
void GiRaFFE_NRPy_RHSs(const paramstruct *restrict params,REAL *restrict auxevol_gfs,const REAL *restrict in_gfs,REAL *restrict rhs_gfs) {
#include "set_Cparameters.h"
// First thing's first: initialize the RHSs to zero!
#pragma omp parallel for
for(int ii=0;ii<Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2*NUM_EVOL_GFS;ii++) {
rhs_gfs[ii] = 0.0;
}
// Next calculate the easier source terms that don't require flux directions
// This will also reset the RHSs for each gf at each new timestep.
calculate_AD_gauge_term_psi6Phi_flux_term_for_RHSs(params,in_gfs,auxevol_gfs);
calculate_AD_gauge_psi6Phi_RHSs(params,in_gfs,auxevol_gfs,rhs_gfs);
// Now, we set up a bunch of structs of pointers to properly guide the PPM algorithm.
// They also count the number of ghostzones available.
gf_and_gz_struct in_prims[NUM_RECONSTRUCT_GFS], out_prims_r[NUM_RECONSTRUCT_GFS], out_prims_l[NUM_RECONSTRUCT_GFS];
int which_prims_to_reconstruct[NUM_RECONSTRUCT_GFS],num_prims_to_reconstruct;
const int Nxxp2NG012 = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;
REAL *temporary = auxevol_gfs + Nxxp2NG012*AEVOLPARENGF; //We're not using this anymore
// This sets pointers to the portion of auxevol_gfs containing the relevant gridfunction.
int ww=0;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAVU0GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_RU0GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_LU0GF;
ww++;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAVU1GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_RU1GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_LU1GF;
ww++;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAVU2GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_RU2GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*VALENCIAV_LU2GF;
ww++;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*BU0GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*B_RU0GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*B_LU0GF;
ww++;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*BU1GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*B_RU1GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*B_LU1GF;
ww++;
in_prims[ww].gf = auxevol_gfs + Nxxp2NG012*BU2GF;
out_prims_r[ww].gf = auxevol_gfs + Nxxp2NG012*B_RU2GF;
out_prims_l[ww].gf = auxevol_gfs + Nxxp2NG012*B_LU2GF;
ww++;
// Prims are defined AT ALL GRIDPOINTS, so we set the # of ghostzones to zero:
for(int i=0;i<NUM_RECONSTRUCT_GFS;i++) for(int j=1;j<=3;j++) { in_prims[i].gz_lo[j]=0; in_prims[i].gz_hi[j]=0; }
// Left/right variables are not yet defined, yet we set the # of gz's to zero by default:
for(int i=0;i<NUM_RECONSTRUCT_GFS;i++) for(int j=1;j<=3;j++) { out_prims_r[i].gz_lo[j]=0; out_prims_r[i].gz_hi[j]=0; }
for(int i=0;i<NUM_RECONSTRUCT_GFS;i++) for(int j=1;j<=3;j++) { out_prims_l[i].gz_lo[j]=0; out_prims_l[i].gz_hi[j]=0; }
ww=0;
which_prims_to_reconstruct[ww]=VX; ww++;
which_prims_to_reconstruct[ww]=VY; ww++;
which_prims_to_reconstruct[ww]=VZ; ww++;
which_prims_to_reconstruct[ww]=BX; ww++;
which_prims_to_reconstruct[ww]=BY; ww++;
which_prims_to_reconstruct[ww]=BZ; ww++;
num_prims_to_reconstruct=ww;
// In each direction, perform the PPM reconstruction procedure.
// Then, add the fluxes to the RHS as appropriate.
for(int flux_dirn=0;flux_dirn<3;flux_dirn++) {
// In each direction, interpolate the metric gfs (gamma,beta,alpha) to cell faces.
interpolate_metric_gfs_to_cell_faces(params,auxevol_gfs,flux_dirn+1);
// Then, reconstruct the primitive variables on the cell faces.
// This function is housed in the file: "reconstruct_set_of_prims_PPM_GRFFE_NRPy.c"
reconstruct_set_of_prims_PPM_GRFFE_NRPy(params, auxevol_gfs, flux_dirn+1, num_prims_to_reconstruct,
which_prims_to_reconstruct, in_prims, out_prims_r, out_prims_l, temporary);
// For example, if flux_dirn==0, then at gamma_faceDD00(i,j,k) represents gamma_{xx}
// at (i-1/2,j,k), Valenciav_lU0(i,j,k) is the x-component of the velocity at (i-1/2-epsilon,j,k),
// and Valenciav_rU0(i,j,k) is the x-component of the velocity at (i-1/2+epsilon,j,k).
if(flux_dirn==0) {
// Next, we calculate the source term for StildeD. Again, this also resets the rhs_gfs array at
// each new timestep.
calculate_StildeD0_source_term(params,auxevol_gfs,rhs_gfs);
// Now, compute the electric field on each face of a cell and add it to the RHSs as appropriate
//calculate_E_field_D0_right(params,auxevol_gfs,rhs_gfs);
//calculate_E_field_D0_left(params,auxevol_gfs,rhs_gfs);
// Finally, we calculate the flux of StildeD and add the appropriate finite-differences
// to the RHSs.
calculate_Stilde_flux_D0(params,auxevol_gfs,rhs_gfs);
}
else if(flux_dirn==1) {
calculate_StildeD1_source_term(params,auxevol_gfs,rhs_gfs);
//calculate_E_field_D1_right(params,auxevol_gfs,rhs_gfs);
//calculate_E_field_D1_left(params,auxevol_gfs,rhs_gfs);
calculate_Stilde_flux_D1(params,auxevol_gfs,rhs_gfs);
}
else {
calculate_StildeD2_source_term(params,auxevol_gfs,rhs_gfs);
//calculate_E_field_D2_right(params,auxevol_gfs,rhs_gfs);
//calculate_E_field_D2_left(params,auxevol_gfs,rhs_gfs);
calculate_Stilde_flux_D2(params,auxevol_gfs,rhs_gfs);
}
for(int count=0;count<=1;count++) {
// This function is written to be general, using notation that matches the forward permutation added to AD2,
// i.e., [F_HLL^x(B^y)]_z corresponding to flux_dirn=0, count=1.
// The SIGN parameter is necessary because
// -E_z(x_i,y_j,z_k) = 0.25 ( [F_HLL^x(B^y)]_z(i+1/2,j,k)+[F_HLL^x(B^y)]_z(i-1/2,j,k)
// -[F_HLL^y(B^x)]_z(i,j+1/2,k)-[F_HLL^y(B^x)]_z(i,j-1/2,k) )
// Note the negative signs on the reversed permutation terms!
// By cyclically permuting with flux_dirn, we
// get contributions to the other components, and by incrementing count, we get the backward permutations:
// Let's suppose flux_dirn = 0. Then we will need to update Ay (count=0) and Az (count=1):
// flux_dirn=count=0 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (0+1+0)%3=AD1GF <- Updating Ay!
// (flux_dirn)%3 = (0)%3 = 0 Vx
// (flux_dirn-count+2)%3 = (0-0+2)%3 = 2 Vz . Inputs Vx, Vz -> SIGN = -1 ; 2.0*((REAL)count)-1.0=-1 check!
// flux_dirn=0,count=1 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (0+1+1)%3=AD2GF <- Updating Az!
// (flux_dirn)%3 = (0)%3 = 0 Vx
// (flux_dirn-count+2)%3 = (0-1+2)%3 = 1 Vy . Inputs Vx, Vy -> SIGN = +1 ; 2.0*((REAL)count)-1.0=2-1=+1 check!
// Let's suppose flux_dirn = 1. Then we will need to update Az (count=0) and Ax (count=1):
// flux_dirn=1,count=0 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (1+1+0)%3=AD2GF <- Updating Az!
// (flux_dirn)%3 = (1)%3 = 1 Vy
// (flux_dirn-count+2)%3 = (1-0+2)%3 = 0 Vx . Inputs Vy, Vx -> SIGN = -1 ; 2.0*((REAL)count)-1.0=-1 check!
// flux_dirn=count=1 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (1+1+1)%3=AD0GF <- Updating Ax!
// (flux_dirn)%3 = (1)%3 = 1 Vy
// (flux_dirn-count+2)%3 = (1-1+2)%3 = 2 Vz . Inputs Vy, Vz -> SIGN = +1 ; 2.0*((REAL)count)-1.0=2-1=+1 check!
// Let's suppose flux_dirn = 2. Then we will need to update Ax (count=0) and Ay (count=1):
// flux_dirn=2,count=0 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (2+1+0)%3=AD0GF <- Updating Ax!
// (flux_dirn)%3 = (2)%3 = 2 Vz
// (flux_dirn-count+2)%3 = (2-0+2)%3 = 1 Vy . Inputs Vz, Vy -> SIGN = -1 ; 2.0*((REAL)count)-1.0=-1 check!
// flux_dirn=2,count=1 -> AD0GF+(flux_dirn+1+count)%3 = AD0GF + (2+1+1)%3=AD1GF <- Updating Ay!
// (flux_dirn)%3 = (2)%3 = 2 Vz
// (flux_dirn-count+2)%3 = (2-1+2)%3 = 0 Vx . Inputs Vz, Vx -> SIGN = +1 ; 2.0*((REAL)count)-1.0=2-1=+1 check!
calculate_E_field_flat_all_in_one(params,
&auxevol_gfs[IDX4ptS(VALENCIAV_RU0GF+(flux_dirn)%3, 0)],&auxevol_gfs[IDX4ptS(VALENCIAV_RU0GF+(flux_dirn-count+2)%3, 0)],
&auxevol_gfs[IDX4ptS(VALENCIAV_LU0GF+(flux_dirn)%3, 0)],&auxevol_gfs[IDX4ptS(VALENCIAV_LU0GF+(flux_dirn-count+2)%3, 0)],
&auxevol_gfs[IDX4ptS(B_RU0GF +(flux_dirn)%3, 0)],&auxevol_gfs[IDX4ptS(B_RU0GF +(flux_dirn-count+2)%3, 0)],
&auxevol_gfs[IDX4ptS(B_LU0GF +(flux_dirn)%3, 0)],&auxevol_gfs[IDX4ptS(B_LU0GF +(flux_dirn-count+2)%3, 0)],
&auxevol_gfs[IDX4ptS(B_RU0GF +(flux_dirn-count+2)%3, 0)],
&auxevol_gfs[IDX4ptS(B_LU0GF +(flux_dirn-count+2)%3, 0)],
&rhs_gfs[IDX4ptS(AD0GF+(flux_dirn+1+count)%3,0)], 2.0*((REAL)count)-1.0, flux_dirn);
}
}
}
void GiRaFFE_NRPy_post_step(const paramstruct *restrict params,REAL *xx[3],REAL *restrict auxevol_gfs,REAL *restrict evol_gfs,const int n) {
// First, apply BCs to AD and psi6Phi. Then calculate BU from AD
apply_bcs_potential(params,evol_gfs);
driver_A_to_B(params,evol_gfs,auxevol_gfs);
//override_BU_with_old_GiRaFFE(params,auxevol_gfs,n);
// Apply fixes to StildeD, then recompute the velocity at the new timestep.
// Apply the current sheet prescription to the velocities
GiRaFFE_NRPy_cons_to_prims(params,xx,auxevol_gfs,evol_gfs);
// Then, recompute StildeD to be consistent with the new velocities
//GiRaFFE_NRPy_prims_to_cons(params,auxevol_gfs,evol_gfs);
// Finally, apply outflow boundary conditions to the velocities.
apply_bcs_velocity(params,auxevol_gfs);
}
""")
def ScalarField_RHSs():
# Step B.4: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step B.5: Import all basic (unrescaled) BSSN scalars & tensors
Bq.BSSN_basic_tensors()
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
Bq.gammabar__inverse_and_derivs()
gammabarUU = Bq.gammabarUU
global sf_rhs, sfM_rhs
# Step B.5.a: Declare grid functions for varphi and Pi
sf, sfM = sfgfs.declare_scalar_field_gridfunctions_if_not_declared_already(
)
# Step 2.a: Add Term 1 to sf_rhs: -alpha*Pi
sf_rhs = -alpha * sfM
# Step 2.b: Add Term 2 to sf_rhs: beta^{i}\partial_{i}\varphi
sf_dupD = ixp.declarerank1("sf_dupD")
for i in range(DIM):
sf_rhs += betaU[i] * sf_dupD[i]
# Step 3a: Add Term 1 to sfM_rhs: alpha * K * Pi
sfM_rhs = alpha * trK * sfM
# Step 3b: Add Term 2 to sfM_rhs: beta^{i}\partial_{i}Pi
sfM_dupD = ixp.declarerank1("sfM_dupD")
for i in range(DIM):
sfM_rhs += betaU[i] * sfM_dupD[i]
# Step 3c: Adding Term 3 to sfM_rhs
# Step 3c.i: Term 3a: gammabar^{ij}\partial_{i}\alpha\partial_{j}\varphi
alpha_dD = ixp.declarerank1("alpha_dD")
sf_dD = ixp.declarerank1("sf_dD")
sfMrhsTerm3 = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -gammabarUU[i][j] * alpha_dD[i] * sf_dD[j]
# Step 3c.ii: Term 3b: \alpha*gammabar^{ij}\partial_{i}\partial_{j}\varphi
sf_dDD = ixp.declarerank2("sf_dDD", "sym01")
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -alpha * gammabarUU[i][j] * sf_dDD[i][j]
# Step 3c.iii: Term 3c: 2*alpha*gammabar^{ij}\partial_{j}\varphi\partial_{i}\phi
Bq.phi_and_derivs(
) # sets exp^{-4phi} = exp_m4phi and \partial_{i}phi = phi_dD[i]
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -2 * alpha * gammabarUU[i][j] * sf_dD[
j] * Bq.phi_dD[i]
# Step 3c.iv: Multiplying Term 3 by e^{-4phi} and adding it to sfM_rhs
sfMrhsTerm3 *= Bq.exp_m4phi
sfM_rhs += sfMrhsTerm3
# Step 3d: Adding Term 4 to sfM_rhs
# Step 3d.i: Term 4a: \alpha \bar\Lambda^{i}\partial_{i}\varphi
LambdabarU = Bq.LambdabarU
sfMrhsTerm4 = sp.sympify(0)
for i in range(DIM):
sfMrhsTerm4 += alpha * LambdabarU[i] * sf_dD[i]
# Step 3d.ii: Evaluating \bar\gamma^{ij}\hat\Gamma^{k}_{ij}
GammahatUDD = rfm.GammahatUDD
gammabarGammahatContractionU = ixp.zerorank1()
for k in range(DIM):
for i in range(DIM):
for j in range(DIM):
gammabarGammahatContractionU[
k] += gammabarUU[i][j] * GammahatUDD[k][i][j]
# Step 3d.iii: Term 4b: \alpha \bar\gamma^{ij}\hat\Gamma^{k}_{ij}\partial_{k}\varphi
for i in range(DIM):
sfMrhsTerm4 += alpha * gammabarGammahatContractionU[i] * sf_dD[i]
# Step 3d.iii: Multplying Term 4 by e^{-4phi} and adding it to sfM_rhs
sfMrhsTerm4 *= Bq.exp_m4phi
sfM_rhs += sfMrhsTerm4
return
def write_dfdr_function(self, Ccodesdir, fd_order=2):
# function to write c code to calculate dfdr term in Sommerfeld boundary condition
# Read what # of dimensions being usded
DIM = par.parval_from_str("grid::DIM")
# Set up the chosen reference metric from chosen coordinate system, set within NRPy+
CoordSystem = par.parval_from_str("reference_metric::CoordSystem")
rfm.reference_metric()
# Simplifying the results make them easier to interpret.
do_simplify = True
if "Sinh" in CoordSystem:
# Simplification takes too long on Sinh* coordinate systems
do_simplify = False
# Construct Jacobian matrix, output Jac_dUSph_dDrfmUD[i][j] = \partial x_{Sph}^i / \partial x^j:
Jac_dUSph_dDrfmUD = ixp.zerorank2()
for i in range(3):
for j in range(3):
Jac_dUSph_dDrfmUD[i][j] = sp.diff(rfm.xxSph[i], rfm.xx[j])
# Invert Jacobian matrix, output to Jac_dUrfm_dDSphUD.
Jac_dUrfm_dDSphUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUSph_dDrfmUD)
# Jac_dUrfm_dDSphUD[i][0] stores \partial x^i / \partial r
if do_simplify:
for i in range(3):
Jac_dUrfm_dDSphUD[i][0] = sp.simplify(Jac_dUrfm_dDSphUD[i][0])
# Declare \partial_i f, which is actually computed later on
fdD = ixp.declarerank1("fdD") # = [fdD0, fdD1, fdD2]
contraction = sp.sympify(0)
for i in range(3):
contraction += fdD[i] * Jac_dUrfm_dDSphUD[i][0]
contraction = sp.simplify(contraction)
r_str_and_contraction_str = outputC([rfm.xxSph[0], contraction],
["*_r", "*_partial_i_f"],
filename="returnstring",
params="includebraces=False")
def gen_central_fd_stencil_str(intdirn, fd_order):
if fd_order == 2:
if intdirn == 0:
return "(gfs[IDX4S(which_gf,i0+1,i1,i2)]-gfs[IDX4S(which_gf,i0-1,i1,i2)])*0.5" # Does not include the 1/dx multiplication
elif intdirn == 1:
return "(gfs[IDX4S(which_gf,i0,i1+1,i2)]-gfs[IDX4S(which_gf,i0,i1-1,i2)])*0.5" # Does not include the 1/dy multiplication
elif intdirn == 2:
return "(gfs[IDX4S(which_gf,i0,i1,i2+1)]-gfs[IDX4S(which_gf,i0,i1,i2-1)])*0.5" # Does not include the 1/dz multiplication
def output_dfdx(intdirn, fd_order):
dirn = str(intdirn)
dirnp1 = str(
(intdirn + 1) % 3
) # if dirn='0', then we want this to be '1'; '1' then '2'; and '2' then '0'
dirnp2 = str(
(intdirn + 2) % 3
) # if dirn='0', then we want this to be '2'; '1' then '0'; and '2' then '1'
if fd_order == 2:
return """
// On a +x""" + dirn + """ or -x""" + dirn + """ face, do up/down winding as appropriate:
if(abs(FACEXi[""" + dirn + """])==1 || i""" + dirn + """+NGHOSTS >= Nxx_plus_2NGHOSTS""" + dirn + """ || i""" + dirn + """-NGHOSTS <= 0) {
int8_t SHIFTSTENCIL""" + dirn + """ = FACEXi[""" + dirn + """];
if(i""" + dirn + """+NGHOSTS >= Nxx_plus_2NGHOSTS""" + dirn + """) SHIFTSTENCIL""" + dirn + """ = -1;
if(i""" + dirn + """-NGHOSTS <= 0) SHIFTSTENCIL""" + dirn + """ = +1;
SHIFTSTENCIL""" + dirnp1 + """ = 0;
SHIFTSTENCIL""" + dirnp2 + """ = 0;
fdD""" + dirn + """
= SHIFTSTENCIL""" + dirn + """*(-1.5*gfs[IDX4S(which_gf,i0+0*SHIFTSTENCIL0,i1+0*SHIFTSTENCIL1,i2+0*SHIFTSTENCIL2)]
+2.*gfs[IDX4S(which_gf,i0+1*SHIFTSTENCIL0,i1+1*SHIFTSTENCIL1,i2+1*SHIFTSTENCIL2)]
-0.5*gfs[IDX4S(which_gf,i0+2*SHIFTSTENCIL0,i1+2*SHIFTSTENCIL1,i2+2*SHIFTSTENCIL2)]
)*invdx""" + dirn + """;
// Not on a +x""" + dirn + """ or -x""" + dirn + """ face, using centered difference:
} else {
fdD""" + dirn + """ = """ + gen_central_fd_stencil_str(
intdirn, 2) + """*invdx""" + dirn + """;
}
"""
else:
print("Error: fd_order = " + str(fd_order) +
" currently unsupported.")
sys.exit(1)
contraction_term_func = """
void contraction_term(const paramstruct *restrict params, const int which_gf, const REAL *restrict gfs, REAL *restrict xx[3],
const int8_t FACEXi[3], const int i0, const int i1, const int i2, REAL *restrict _r, REAL *restrict _partial_i_f) {
#include "RELATIVE_PATH__set_Cparameters.h" /* Header file containing correct #include for set_Cparameters.h;
* accounting for the relative path */
// Initialize derivatives to crazy values, to ensure that
// we will notice in case they aren't set properly.
REAL fdD0=1e100;
REAL fdD1=1e100;
REAL fdD2=1e100;
REAL xx0 = xx[0][i0];
REAL xx1 = xx[1][i1];
REAL xx2 = xx[2][i2];
int8_t SHIFTSTENCIL0;
int8_t SHIFTSTENCIL1;
int8_t SHIFTSTENCIL2;
"""
for i in range(DIM):
if "fdD" + str(i) in r_str_and_contraction_str:
contraction_term_func += output_dfdx(i, fd_order)
contraction_term_func += "\n" + r_str_and_contraction_str
contraction_term_func += """
} // END contraction_term function
"""
with open(
os.path.join(Ccodesdir,
"boundary_conditions/radial_derivative.h"),
"w") as file:
file.write(contraction_term_func)
def GiRaFFE_NRPy_Afield_flux(Ccodesdir):
cmd.mkdir(Ccodesdir)
# Write out the code to a file.
gammaDD = ixp.declarerank2("gammaDD", "sym01", DIM=3)
betaU = ixp.declarerank1("betaU", DIM=3)
alpha = sp.sympify("alpha")
for flux_dirn in range(3):
chsp.find_cmax_cmin(flux_dirn, gammaDD, betaU, alpha)
Ccode_kernel = outputC([chsp.cmax, chsp.cmin], ["cmax", "cmin"],
"returnstring",
params="outCverbose=False,CSE_sorting=none")
Ccode_kernel = Ccode_kernel.replace("cmax",
"*cmax").replace("cmin", "*cmin")
Ccode_kernel = Ccode_kernel.replace("betaU0", "betaUi").replace(
"betaU1", "betaUi").replace("betaU2", "betaUi")
with open(
os.path.join(Ccodesdir,
"compute_cmax_cmin_dirn" + str(flux_dirn) + ".h"),
"w") as file:
file.write(Ccode_kernel)
with open(os.path.join(Ccodesdir, "calculate_E_field_flat_all_in_one.h"),
"w") as file:
file.write(
r"""void find_cmax_cmin(const REAL gammaDD00, const REAL gammaDD01, const REAL gammaDD02,
const REAL gammaDD11, const REAL gammaDD12, const REAL gammaDD22,
const REAL betaUi, const REAL alpha, const int flux_dirn,
REAL *cmax, REAL *cmin) {
switch(flux_dirn) {
case 0:
#include "compute_cmax_cmin_dirn0.h"
break;
case 1:
#include "compute_cmax_cmin_dirn1.h"
break;
case 2:
#include "compute_cmax_cmin_dirn2.h"
break;
default:
printf("Invalid parameter flux_dirn!"); *cmax = 1.0/0.0; *cmin = 1.0/0.0;
break;
}
}
REAL HLLE_solve(REAL F0B1_r, REAL F0B1_l, REAL U_r, REAL U_l, REAL cmin, REAL cmax) {
// Eq. 3.15 of https://epubs.siam.org/doi/abs/10.1137/1025002?journalCode=siread
// F_HLLE = (c_min F_R + c_max F_L - c_min c_max (U_R-U_L)) / (c_min + c_max)
return (cmin*F0B1_r + cmax*F0B1_l - cmin*cmax*(U_r-U_l)) / (cmin+cmax);
}
/*
Calculate the electric flux on both faces in the input direction.
The input count is an integer that is either 0 or 1. If it is 0, this implies
that the components are input in order of a backwards permutation and the final
results will need to be multiplied by -1.0. If it is 1, then the permutation is forwards.
*/
void calculate_E_field_flat_all_in_one(const paramstruct *params,
const REAL *Vr0,const REAL *Vr1,
const REAL *Vl0,const REAL *Vl1,
const REAL *Br0,const REAL *Br1,
const REAL *Bl0,const REAL *Bl1,
const REAL *Brflux_dirn,
const REAL *Blflux_dirn,
const REAL *gamma_faceDD00, const REAL *gamma_faceDD01, const REAL *gamma_faceDD02,
const REAL *gamma_faceDD11, const REAL *gamma_faceDD12, const REAL *gamma_faceDD22,
const REAL *beta_faceU0, const REAL *beta_faceU1, const REAL *alpha_face,
REAL *A2_rhs,const REAL SIGN,const int flux_dirn) {
// This function is written to be generic and compute the contribution for all three AD RHSs.
// However, for convenience, the notation used in the function itself is for the contribution
// to AD2, specifically the [F_HLL^x(B^y)]_z term, with reconstructions in the x direction. This
// corresponds to flux_dirn=0 and count=1 (which corresponds to SIGN=+1.0).
// Thus, Az(i,j,k) += 0.25 ( [F_HLL^x(B^y)]_z(i+1/2,j,k)+[F_HLL^x(B^y)]_z(i-1/2,j,k)) are solved here.
// The other terms are computed by cyclically permuting the indices when calling this function.
#include "../set_Cparameters.h"
#pragma omp parallel for
for(int i2=NGHOSTS; i2<NGHOSTS+Nxx2; i2++) {
for(int i1=NGHOSTS; i1<NGHOSTS+Nxx1; i1++) {
for(int i0=NGHOSTS; i0<NGHOSTS+Nxx0; i0++) {
// First, we set the index from which we will read memory. indexp1 is incremented by
// one point in the direction of reconstruction. These correspond to the faces at at
// i-1/2 and i+1/2, respectively.
// Now, we read in memory. We need the x and y components of velocity and magnetic field on both
// the left and right sides of the interface at *both* faces.
// Here, the point (i0,i1,i2) corresponds to the point (i-1/2,j,k)
const int index = IDX3S(i0,i1,i2);
const double alpha = alpha_face[index];
const double betaU0 = beta_faceU0[index];
const double betaU1 = beta_faceU1[index];
const double v_rU0 = alpha*Vr0[index]-betaU0;
const double v_rU1 = alpha*Vr1[index]-betaU1;
const double B_rU0 = Br0[index];
const double B_rU1 = Br1[index];
const double B_rflux_dirn = Brflux_dirn[index];
const double v_lU0 = alpha*Vl0[index]-betaU0;
const double v_lU1 = alpha*Vl1[index]-betaU1;
const double B_lU0 = Bl0[index];
const double B_lU1 = Bl1[index];
const double B_lflux_dirn = Blflux_dirn[index];
// We will also need need the square root of the metric determinant here at this point:
const REAL gxx = gamma_faceDD00[index];
const REAL gxy = gamma_faceDD01[index];
const REAL gxz = gamma_faceDD02[index];
const REAL gyy = gamma_faceDD11[index];
const REAL gyz = gamma_faceDD12[index];
const REAL gzz = gamma_faceDD22[index];
const REAL sqrtgammaDET = sqrt( gxx*gyy*gzz
- gxx*gyz*gyz
+2*gxy*gxz*gyz
- gyy*gxz*gxz
- gzz*gxy*gxy );
// *******************************
// REPEAT ABOVE, but at i+1, which corresponds to point (i+1/2,j,k)
// Recall that the documentation here assumes flux_dirn==0, but the
// algorithm is generalized so that any flux_dirn or velocity/magnetic
// field component can be computed via permuting the inputs into this
// function.
const int indexp1 = IDX3S(i0+(flux_dirn==0),i1+(flux_dirn==1),i2+(flux_dirn==2));
const double alpha_p1 = alpha_face[indexp1];
const double betaU0_p1 = beta_faceU0[indexp1];
const double betaU1_p1 = beta_faceU1[indexp1];
const double v_rU0_p1 = alpha_p1*Vr0[indexp1]-betaU0_p1;
const double v_rU1_p1 = alpha_p1*Vr1[indexp1]-betaU1_p1;
const double B_rU0_p1 = Br0[indexp1];
const double B_rU1_p1 = Br1[indexp1];
const double B_rflux_dirn_p1 = Brflux_dirn[indexp1];
const double v_lU0_p1 = alpha_p1*Vl0[indexp1]-betaU0_p1;
const double v_lU1_p1 = alpha_p1*Vl1[indexp1]-betaU1_p1;
const double B_lU0_p1 = Bl0[indexp1];
const double B_lU1_p1 = Bl1[indexp1];
const double B_lflux_dirn_p1 = Blflux_dirn[indexp1];
// We will also need need the square root of the metric determinant here at this point:
const REAL gxx_p1 = gamma_faceDD00[indexp1];
const REAL gxy_p1 = gamma_faceDD01[indexp1];
const REAL gxz_p1 = gamma_faceDD02[indexp1];
const REAL gyy_p1 = gamma_faceDD11[indexp1];
const REAL gyz_p1 = gamma_faceDD12[indexp1];
const REAL gzz_p1 = gamma_faceDD22[indexp1];
const REAL sqrtgammaDET_p1 = sqrt( gxx_p1*gyy_p1*gzz_p1
- gxx_p1*gyz_p1*gyz_p1
+2*gxy_p1*gxz_p1*gyz_p1
- gyy_p1*gxz_p1*gxz_p1
- gzz_p1*gxy_p1*gxy_p1 );
// *******************************
// DEBUGGING:
// if(flux_dirn==0 && SIGN>0 && i1==Nxx_plus_2NGHOSTS1/2 && i2==Nxx_plus_2NGHOSTS2/2) {
// printf("index=%d & indexp1=%d\n",index,indexp1);
// }
// Since we are computing A_z, the relevant equation here is:
// -E_z(x_i,y_j,z_k) = 0.25 ( [F_HLL^x(B^y)]_z(i+1/2,j,k)+[F_HLL^x(B^y)]_z(i-1/2,j,k)
// -[F_HLL^y(B^x)]_z(i,j+1/2,k)-[F_HLL^y(B^x)]_z(i,j-1/2,k) )
// We will construct the above sum one half at a time, first with SIGN=+1, which
// corresponds to flux_dirn = 0, count=1, and
// takes care of the terms:
// [F_HLL^x(B^y)]_z(i+1/2,j,k)+[F_HLL^x(B^y)]_z(i-1/2,j,k)
// ( Note that we will repeat the above with flux_dirn = 1, count = 0, with SIGN=-1
// AND with the input components switched (x->y,y->x) so that we get the term
// -[F_HLL^y(B^x)]_z(i,j+1/2,k)-[F_HLL^y(B^x)]_z(i,j-1/2,k)
// thus completing the above sum. )
// Here, [F_HLL^i(B^j)]_k = (v^i B^j - v^j B^i) in general.
// Calculate the flux vector on each face for each component of the E-field:
// The F(B) terms are as Eq. 6 in Giacomazzo: https://arxiv.org/pdf/1009.2468.pdf
// [F^i(B^j)]_k = \sqrt{\gamma} (v^i B^j - v^j B^i)
// Therefore since we want [F_HLL^x(B^y)]_z,
// we will code (v^x B^y - v^y B^x) on both left and right faces.
const REAL F0B1_r = sqrtgammaDET*(v_rU0*B_rU1 - v_rU1*B_rU0);
const REAL F0B1_l = sqrtgammaDET*(v_lU0*B_lU1 - v_lU1*B_lU0);
// Compute the state vector for these terms:
const REAL U_r = B_rflux_dirn;
const REAL U_l = B_lflux_dirn;
REAL cmin,cmax;
// Basic HLLE solver:
find_cmax_cmin(gxx,gxy,gxz,
gyy,gyz,gzz,
betaU0,alpha,flux_dirn,
&cmax, &cmin);
const REAL FHLL_0B1 = HLLE_solve(F0B1_r, F0B1_l, U_r, U_l, cmin, cmax);
// ************************************
// ************************************
// REPEAT ABOVE, but at point i+1
// Calculate the flux vector on each face for each component of the E-field:
const REAL F0B1_r_p1 = sqrtgammaDET_p1*(v_rU0_p1*B_rU1_p1 - v_rU1_p1*B_rU0_p1);
const REAL F0B1_l_p1 = sqrtgammaDET_p1*(v_lU0_p1*B_lU1_p1 - v_lU1_p1*B_lU0_p1);
// Compute the state vector for this flux direction
const REAL U_r_p1 = B_rflux_dirn_p1;
const REAL U_l_p1 = B_lflux_dirn_p1;
//const REAL U_r_p1 = B_rU1_p1;
//const REAL U_l_p1 = B_lU1_p1;
// Basic HLLE solver, but at the next point:
find_cmax_cmin(gxx_p1,gxy_p1,gxz_p1,
gyy_p1,gyz_p1,gzz_p1,
betaU0_p1,alpha_p1,flux_dirn,
&cmax, &cmin);
const REAL FHLL_0B1p1 = HLLE_solve(F0B1_r_p1, F0B1_l_p1, U_r_p1, U_l_p1, cmin, cmax);
// ************************************
// ************************************
// With the Riemann problem solved, we add the contributions to the RHSs:
// -E_z(x_i,y_j,z_k) &= 0.25 ( [F_HLL^x(B^y)]_z(i+1/2,j,k)+[F_HLL^x(B^y)]_z(i-1/2,j,k)
// -[F_HLL^y(B^x)]_z(i,j+1/2,k)-[F_HLL^y(B^x)]_z(i,j-1/2,k) )
// (Eq. 11 in https://arxiv.org/pdf/1009.2468.pdf)
// This code, as written, solves the first two terms for flux_dirn=0. Calling this function for count=0
// and flux_dirn=1 flips x for y to solve the latter two, switching to SIGN=-1 as well.
// Here, we finally add together the output of the HLLE solver at i-1/2 and i+1/2
// We also multiply by the SIGN dictated by the order of the input vectors and divide by 4.
A2_rhs[index] += SIGN*0.25*(FHLL_0B1 + FHLL_0B1p1);
// flux dirn = 0 ===================> i-1/2 i+1/2
// Eq 11 in Giacomazzo:
// -FxBy(avg over i-1/2 and i+1/2) + FyBx(avg over j-1/2 and j+1/2)
// Eq 6 in Giacomazzo:
// FxBy = vxBy - vyBx
// ->
// FHLL_0B1 = vyBx - vxBy
} // END LOOP: for(int i0=NGHOSTS; i0<NGHOSTS+Nxx0; i0++)
} // END LOOP: for(int i1=NGHOSTS; i1<NGHOSTS+Nxx1; i1++)
} // END LOOP: for(int i2=NGHOSTS; i2<NGHOSTS+Nxx2; i2++)
}
""")
def BSSN_RHSs__generate_symbolic_expressions():
######################################
# START: GENERATE SYMBOLIC EXPRESSIONS
print("Generating symbolic expressions for BSSN RHSs...")
start = time.time()
# Enable rfm_precompute infrastructure, which results in
# BSSN RHSs that are free of transcendental functions,
# even in curvilinear coordinates, so long as
# ConformalFactor is set to "W" (default).
par.set_parval_from_str("reference_metric::enable_rfm_precompute","True")
par.set_parval_from_str("reference_metric::rfm_precompute_Ccode_outdir",os.path.join(outdir,"rfm_files/"))
# Evaluate BSSN + BSSN gauge RHSs with rfm_precompute enabled:
import BSSN.BSSN_quantities as Bq
par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic","True")
rhs.BSSN_RHSs()
if T4UU != None:
import BSSN.BSSN_stress_energy_source_terms as Bsest
Bsest.BSSN_source_terms_for_BSSN_RHSs(T4UU)
rhs.trK_rhs += Bsest.sourceterm_trK_rhs
for i in range(3):
# Needed for Gamma-driving shift RHSs:
rhs.Lambdabar_rhsU[i] += Bsest.sourceterm_Lambdabar_rhsU[i]
# Needed for BSSN RHSs:
rhs.lambda_rhsU[i] += Bsest.sourceterm_lambda_rhsU[i]
for j in range(3):
rhs.a_rhsDD[i][j] += Bsest.sourceterm_a_rhsDD[i][j]
gaugerhs.BSSN_gauge_RHSs()
# Add Kreiss-Oliger dissipation to the BSSN RHSs:
thismodule = "KO_Dissipation"
diss_strength = par.Cparameters("REAL", thismodule, "diss_strength", default_KO_strength)
alpha_dKOD = ixp.declarerank1("alpha_dKOD")
cf_dKOD = ixp.declarerank1("cf_dKOD")
trK_dKOD = ixp.declarerank1("trK_dKOD")
betU_dKOD = ixp.declarerank2("betU_dKOD","nosym")
vetU_dKOD = ixp.declarerank2("vetU_dKOD","nosym")
lambdaU_dKOD = ixp.declarerank2("lambdaU_dKOD","nosym")
aDD_dKOD = ixp.declarerank3("aDD_dKOD","sym01")
hDD_dKOD = ixp.declarerank3("hDD_dKOD","sym01")
for k in range(3):
gaugerhs.alpha_rhs += diss_strength*alpha_dKOD[k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
rhs.cf_rhs += diss_strength* cf_dKOD[k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
rhs.trK_rhs += diss_strength* trK_dKOD[k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
for i in range(3):
if "2ndOrder" in ShiftCondition:
gaugerhs.bet_rhsU[i] += diss_strength* betU_dKOD[i][k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
gaugerhs.vet_rhsU[i] += diss_strength* vetU_dKOD[i][k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
rhs.lambda_rhsU[i] += diss_strength*lambdaU_dKOD[i][k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
for j in range(3):
rhs.a_rhsDD[i][j] += diss_strength*aDD_dKOD[i][j][k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
rhs.h_rhsDD[i][j] += diss_strength*hDD_dKOD[i][j][k]*rfm.ReU[k] # ReU[k] = 1/scalefactor_orthog_funcform[k]
# We use betaU as our upwinding control vector:
Bq.BSSN_basic_tensors()
betaU = Bq.betaU
# Now that we are finished with all the rfm hatted
# quantities in generic precomputed functional
# form, let's restore them to their closed-
# form expressions.
par.set_parval_from_str("reference_metric::enable_rfm_precompute","False") # Reset to False to disable rfm_precompute.
rfm.ref_metric__hatted_quantities()
par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic","False")
end = time.time()
print("(BENCH) Finished BSSN RHS symbolic expressions in "+str(end-start)+" seconds.")
# END: GENERATE SYMBOLIC EXPRESSIONS
######################################
BSSN_RHSs_SymbExpressions = [lhrh(lhs=gri.gfaccess("rhs_gfs","aDD00"), rhs=rhs.a_rhsDD[0][0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","aDD01"), rhs=rhs.a_rhsDD[0][1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","aDD02"), rhs=rhs.a_rhsDD[0][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","aDD11"), rhs=rhs.a_rhsDD[1][1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","aDD12"), rhs=rhs.a_rhsDD[1][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","aDD22"), rhs=rhs.a_rhsDD[2][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","alpha"), rhs=gaugerhs.alpha_rhs),
lhrh(lhs=gri.gfaccess("rhs_gfs","betU0"), rhs=gaugerhs.bet_rhsU[0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","betU1"), rhs=gaugerhs.bet_rhsU[1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","betU2"), rhs=gaugerhs.bet_rhsU[2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","cf"), rhs=rhs.cf_rhs),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD00"), rhs=rhs.h_rhsDD[0][0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD01") ,rhs=rhs.h_rhsDD[0][1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD02"), rhs=rhs.h_rhsDD[0][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD11"), rhs=rhs.h_rhsDD[1][1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD12"), rhs=rhs.h_rhsDD[1][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","hDD22"), rhs=rhs.h_rhsDD[2][2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","lambdaU0"),rhs=rhs.lambda_rhsU[0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","lambdaU1"),rhs=rhs.lambda_rhsU[1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","lambdaU2"),rhs=rhs.lambda_rhsU[2]),
lhrh(lhs=gri.gfaccess("rhs_gfs","trK"), rhs=rhs.trK_rhs),
lhrh(lhs=gri.gfaccess("rhs_gfs","vetU0"), rhs=gaugerhs.vet_rhsU[0]),
lhrh(lhs=gri.gfaccess("rhs_gfs","vetU1"), rhs=gaugerhs.vet_rhsU[1]),
lhrh(lhs=gri.gfaccess("rhs_gfs","vetU2"), rhs=gaugerhs.vet_rhsU[2]) ]
return [betaU,BSSN_RHSs_SymbExpressions]
def ADM_in_terms_of_BSSN():
global gammaDD, gammaDDdD, gammaDDdDD, gammaUU, detgamma, GammaUDD, KDD, KDDdD
# Step 1.c: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 1.d: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.e: Import all basic (unrescaled) BSSN scalars & tensors
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors()
gammabarDD = Bq.gammabarDD
cf = Bq.cf
AbarDD = Bq.AbarDD
trK = Bq.trK
Bq.gammabar__inverse_and_derivs()
gammabarDD_dD = Bq.gammabarDD_dD
gammabarDD_dDD = Bq.gammabarDD_dDD
Bq.AbarUU_AbarUD_trAbar_AbarDD_dD()
AbarDD_dD = Bq.AbarDD_dD
# Step 2: The ADM three-metric gammaDD and its
# derivatives in terms of BSSN quantities.
gammaDD = ixp.zerorank2()
exp4phi = sp.sympify(0)
if par.parval_from_str("EvolvedConformalFactor_cf") == "phi":
exp4phi = sp.exp(4 * cf)
elif par.parval_from_str("EvolvedConformalFactor_cf") == "chi":
exp4phi = (1 / cf)
elif par.parval_from_str("EvolvedConformalFactor_cf") == "W":
exp4phi = (1 / cf ** 2)
else:
print("Error EvolvedConformalFactor_cf type = \"" + par.parval_from_str("EvolvedConformalFactor_cf") + "\" unknown.")
sys.exit(1)
for i in range(DIM):
for j in range(DIM):
gammaDD[i][j] = exp4phi * gammabarDD[i][j]
# Step 2.a: Derivatives of $e^{4\phi}$
phidD = ixp.zerorank1()
phidDD = ixp.zerorank2()
cf_dD = ixp.declarerank1("cf_dD")
cf_dDD = ixp.declarerank2("cf_dDD","sym01")
if par.parval_from_str("EvolvedConformalFactor_cf") == "phi":
for i in range(DIM):
phidD[i] = cf_dD[i]
for j in range(DIM):
phidDD[i][j] = cf_dDD[i][j]
elif par.parval_from_str("EvolvedConformalFactor_cf") == "chi":
for i in range(DIM):
phidD[i] = -sp.Rational(1,4)*exp4phi*cf_dD[i]
for j in range(DIM):
phidDD[i][j] = sp.Rational(1,4)*( exp4phi**2*cf_dD[i]*cf_dD[j] - exp4phi*cf_dDD[i][j] )
elif par.parval_from_str("EvolvedConformalFactor_cf") == "W":
exp2phi = (1 / cf)
for i in range(DIM):
phidD[i] = -sp.Rational(1,2)*exp2phi*cf_dD[i]
for j in range(DIM):
phidDD[i][j] = sp.Rational(1,2)*( exp4phi*cf_dD[i]*cf_dD[j] - exp2phi*cf_dDD[i][j] )
else:
print("Error EvolvedConformalFactor_cf type = \""+par.parval_from_str("EvolvedConformalFactor_cf")+"\" unknown.")
sys.exit(1)
exp4phidD = ixp.zerorank1()
exp4phidDD = ixp.zerorank2()
for i in range(DIM):
exp4phidD[i] = 4*exp4phi*phidD[i]
for j in range(DIM):
exp4phidDD[i][j] = 16*exp4phi*phidD[i]*phidD[j] + 4*exp4phi*phidDD[i][j]
# Step 2.b: Derivatives of gammaDD, the ADM three-metric
gammaDDdD = ixp.zerorank3()
gammaDDdDD = ixp.zerorank4()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
gammaDDdD[i][j][k] = exp4phidD[k] * gammabarDD[i][j] + exp4phi * gammabarDD_dD[i][j][k]
for l in range(DIM):
gammaDDdDD[i][j][k][l] = exp4phidDD[k][l] * gammabarDD[i][j] + \
exp4phidD[k] * gammabarDD_dD[i][j][l] + \
exp4phidD[l] * gammabarDD_dD[i][j][k] + \
exp4phi * gammabarDD_dDD[i][j][k][l]
# Step 2.c: 3-Christoffel symbols associated with ADM 3-metric gammaDD
# Step 2.c.i: First compute the inverse 3-metric gammaUU:
gammaUU, detgamma = ixp.symm_matrix_inverter3x3(gammaDD)
GammaUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GammaUDD[i][j][k] += sp.Rational(1,2)*gammaUU[i][l]* \
(gammaDDdD[l][j][k] + gammaDDdD[l][k][j] - gammaDDdD[j][k][l])
# Step 3: Define ADM extrinsic curvature KDD and
# its first spatial derivatives KDDdD
# in terms of BSSN quantities
KDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
KDD[i][j] = exp4phi * AbarDD[i][j] + sp.Rational(1, 3) * gammaDD[i][j] * trK
KDDdD = ixp.zerorank3()
trK_dD = ixp.declarerank1("trK_dD")
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
KDDdD[i][j][k] = exp4phidD[k] * AbarDD[i][j] + exp4phi * AbarDD_dD[i][j][k] + \
sp.Rational(1, 3) * (gammaDDdD[i][j][k] * trK + gammaDD[i][j] * trK_dD[k])
def GiRaFFEfood_NRPy_Exact_Wald():
# <a id='step2'></a>
#
# ### Step 2: Set the vectors A and E in Spherical coordinates
# $$\label{step2}$$
#
# \[Back to [top](#top)\]
#
# We will first build the fundamental vectors $A_i$ and $E_i$ in spherical coordinates (see [Table 3](https://arxiv.org/pdf/1704.00599.pdf)). Note that we use reference_metric.py to set $r$ and $\theta$ in terms of Cartesian coordinates; this will save us a step later when we convert to Cartesian coordinates. Since $C_0 = 1$,
# \begin{align}
# A_{\phi} &= \frac{1}{2} r^2 \sin^2 \theta \\
# E_{\phi} &= 2 M \left( 1+ \frac {2M}{r} \right)^{-1/2} \sin^2 \theta. \\
# \end{align}
# While we have $E_i$ set as a variable in NRPy+, note that the final C code won't store these values.
# Step 2: Set the vectors A and E in Spherical coordinates
r = rfm.xxSph[
0] + KerrSchild_radial_shift # We are setting the data up in Shifted Kerr-Schild coordinates
theta = rfm.xxSph[1]
# Initialize all components of A and E in the *spherical basis* to zero
ASphD = ixp.zerorank1()
ESphD = ixp.zerorank1()
ASphD[2] = (r * r * sp.sin(theta)**2) / 2
ESphD[2] = 2 * M * sp.sin(theta)**2 / sp.sqrt(1 + 2 * M / r)
# <a id='step3'></a>
#
# ### Step 3: Use the Jacobian matrix to transform the vectors to Cartesian coordinates.
# $$\label{step3}$$
#
# \[Back to [top](#top)\]
#
# Now, we will use the coordinate transformation definitions provided by reference_metric.py to build the Jacobian
# $$
# \frac{\partial x_{\rm Sph}^j}{\partial x_{\rm Cart}^i},
# $$
# where $x_{\rm Sph}^j \in \{r,\theta,\phi\}$ and $x_{\rm Cart}^i \in \{x,y,z\}$. We would normally compute its inverse, but since none of the quantities we need to transform have upper indices, it is not necessary. Then, since both $A_i$ and $E_i$ have one lower index, both will need to be multiplied by the Jacobian:
#
# $$
# A_i^{\rm Cart} = A_j^{\rm Sph} \frac{\partial x_{\rm Sph}^j}{\partial x_{\rm Cart}^i},
# $$
# Step 3: Use the Jacobian matrix to transform the vectors to Cartesian coordinates.
drrefmetric__dx_0UDmatrix = sp.Matrix([[
sp.diff(rfm.xxSph[0], rfm.xx[0]),
sp.diff(rfm.xxSph[0], rfm.xx[1]),
sp.diff(rfm.xxSph[0], rfm.xx[2])
],
[
sp.diff(rfm.xxSph[1],
rfm.xx[0]),
sp.diff(rfm.xxSph[1],
rfm.xx[1]),
sp.diff(rfm.xxSph[1], rfm.xx[2])
],
[
sp.diff(rfm.xxSph[2],
rfm.xx[0]),
sp.diff(rfm.xxSph[2],
rfm.xx[1]),
sp.diff(rfm.xxSph[2], rfm.xx[2])
]])
#dx__drrefmetric_0UDmatrix = drrefmetric__dx_0UDmatrix.inv() # We don't actually need this in this case.
global AD
AD = ixp.register_gridfunctions_for_single_rank1("EVOL", "AD")
ED = ixp.zerorank1()
for i in range(3):
for j in range(3):
AD[i] = drrefmetric__dx_0UDmatrix[(j, i)] * ASphD[j]
ED[i] = drrefmetric__dx_0UDmatrix[(j, i)] * ESphD[j]
#Step 4: Declare the basic spacetime quantities
alpha = sp.symbols("alpha", real=True)
betaU = ixp.declarerank1("betaU", DIM=3)
gammaDD = ixp.declarerank2("gammaDD", "sym01", DIM=3)
import GRHD.equations as GRHD
GRHD.compute_sqrtgammaDET(gammaDD)
# <a id='step4'></a>
#
# ### Step 4: Calculate $v^i$ from $A_i$ and $E_i$
# $$\label{step4}$$
#
# \[Back to [top](#top)\]
#
# We will now find the magnetic field using equation 18 in [the original paper](https://arxiv.org/pdf/1704.00599.pdf) $$B^i = \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k. $$ We will need the metric quantites: the lapse $\alpha$, the shift $\beta^i$, and the three-metric $\gamma_{ij}$. We will also need the Levi-Civita symbol, provided by $\text{WeylScal4NRPy}$.
# Step 4: Calculate v^i from A_i and E_i
# Step 4a: Calculate the magnetic field B^i
# Here, we build the Levi-Civita tensor from the Levi-Civita symbol.
# Here, we build the Levi-Civita tensor from the Levi-Civita symbol.
import WeylScal4NRPy.WeylScalars_Cartesian as weyl
LeviCivitaSymbolDDD = weyl.define_LeviCivitaSymbol_rank3()
LeviCivitaTensorUUU = ixp.zerorank3()
for i in range(3):
for j in range(3):
for k in range(3):
LeviCivitaTensorUUU[i][j][
k] = LeviCivitaSymbolDDD[i][j][k] / GRHD.sqrtgammaDET
# For the initial data, we can analytically take the derivatives of A_i
ADdD = ixp.zerorank2()
for i in range(3):
for j in range(3):
ADdD[i][j] = sp.simplify(sp.diff(AD[i], rfm.xxCart[j]))
#global BU
BU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
BU[i] += LeviCivitaTensorUUU[i][j][k] * ADdD[k][j]
# We will now build the initial velocity using equation 152 in [this paper,](https://arxiv.org/pdf/1310.3274v2.pdf) cited in the original $\texttt{GiRaFFE}$ code: $$ v^i = \alpha \frac{\epsilon^{ijk} E_j B_k}{B^2} -\beta^i. $$
# However, our code needs the Valencia 3-velocity while this expression is for the drift velocity. So, we will need to transform it to the Valencia 3-velocity using the rule $\bar{v}^i = \frac{1}{\alpha} \left(v^i +\beta^i \right)$.
# Thus, $$\bar{v}^i = \frac{\epsilon^{ijk} E_j B_k}{B^2}$$
# Step 4b: Calculate B^2 and B_i
# B^2 is an inner product defined in the usual way:
B2 = sp.sympify(0)
for i in range(3):
for j in range(3):
B2 += gammaDD[i][j] * BU[i] * BU[j]
# Lower the index on B^i
BD = ixp.zerorank1()
for i in range(3):
for j in range(3):
BD[i] += gammaDD[i][j] * BU[j]
# Step 4c: Calculate the Valencia 3-velocity
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaTensorUUU[i][j][k] * ED[j] * BD[k] / B2
def BSSN_constraints(add_T4UUmunu_source_terms=False):
# Step 1.a: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.b: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 2: Hamiltonian constraint.
# First declare all needed variables
Bq.declare_BSSN_gridfunctions_if_not_declared_already() # Sets trK
Bq.BSSN_basic_tensors() # Sets AbarDD
Bq.gammabar__inverse_and_derivs() # Sets gammabarUU
Bq.AbarUU_AbarUD_trAbar_AbarDD_dD() # Sets AbarUU and AbarDD_dD
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU() # Sets RbarDD
Bq.phi_and_derivs() # Sets phi_dBarD & phi_dBarDD
###############################
###############################
# HAMILTONIAN CONSTRAINT
###############################
###############################
# Term 1: 2/3 K^2
global H
H = sp.Rational(2, 3) * Bq.trK**2
# Term 2: -A_{ij} A^{ij}
for i in range(DIM):
for j in range(DIM):
H += -Bq.AbarDD[i][j] * Bq.AbarUU[i][j]
# Term 3a: trace(Rbar)
Rbartrace = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
Rbartrace += Bq.gammabarUU[i][j] * Bq.RbarDD[i][j]
# Term 3b: -8 \bar{\gamma}^{ij} \bar{D}_i \phi \bar{D}_j \phi = -8*phi_dBar_times_phi_dBar
# Term 3c: -8 \bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \phi = -8*phi_dBarDD_contraction
phi_dBar_times_phi_dBar = sp.sympify(0) # Term 3b
phi_dBarDD_contraction = sp.sympify(0) # Term 3c
for i in range(DIM):
for j in range(DIM):
phi_dBar_times_phi_dBar += Bq.gammabarUU[i][j] * Bq.phi_dBarD[
i] * Bq.phi_dBarD[j]
phi_dBarDD_contraction += Bq.gammabarUU[i][j] * Bq.phi_dBarDD[i][j]
# Add Term 3:
H += Bq.exp_m4phi * (Rbartrace - 8 *
(phi_dBar_times_phi_dBar + phi_dBarDD_contraction))
if add_T4UUmunu_source_terms:
M_PI = par.Cparameters("#define", thismodule, "M_PI",
"") # M_PI is pi as defined in C
BTmunu.define_BSSN_T4UUmunu_rescaled_source_terms()
rho = BTmunu.rho
H += -16 * M_PI * rho
# FIXME: ADD T4UUmunu SOURCE TERMS TO MOMENTUM CONSTRAINT!
# Step 3: M^i, the momentum constraint
###############################
###############################
# MOMENTUM CONSTRAINT
###############################
###############################
# SEE Tutorial-BSSN_constraints.ipynb for full documentation.
global MU
MU = ixp.zerorank1()
# Term 2: 6 A^{ij} \partial_j \phi:
for i in range(DIM):
for j in range(DIM):
MU[i] += 6 * Bq.AbarUU[i][j] * Bq.phi_dD[j]
# Term 3: -2/3 \bar{\gamma}^{ij} K_{,j}
trK_dD = ixp.declarerank1(
"trK_dD") # Not defined in BSSN_RHSs; only trK_dupD is defined there.
for i in range(DIM):
for j in range(DIM):
MU[i] += -sp.Rational(2, 3) * Bq.gammabarUU[i][j] * trK_dD[j]
# First define aDD_dD:
aDD_dD = ixp.declarerank3("aDD_dD", "sym01")
# Then evaluate the conformal covariant derivative \bar{D}_j \bar{A}_{lm}
AbarDD_dBarD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
AbarDD_dBarD[i][j][k] = Bq.AbarDD_dD[i][j][k]
for l in range(DIM):
AbarDD_dBarD[i][j][
k] += -Bq.GammabarUDD[l][k][i] * Bq.AbarDD[l][j]
AbarDD_dBarD[i][j][
k] += -Bq.GammabarUDD[l][k][j] * Bq.AbarDD[i][l]
# Term 1: Contract twice with the metric to make \bar{D}_{j} \bar{A}^{ij}
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
MU[i] += Bq.gammabarUU[i][k] * Bq.gammabarUU[j][
l] * AbarDD_dBarD[k][l][j]
# Finally, we multiply by e^{-4 phi} and rescale the momentum constraint:
for i in range(DIM):
MU[i] *= Bq.exp_m4phi / rfm.ReU[i]
def VacuumMaxwellRHSs_rescaled():
global erhsU, arhsU, psi_rhs, Gamma_rhs, C, G, EU_Cart, AU_Cart
#Step 0: Set the spatial dimension parameter to 3.
par.set_parval_from_str("grid::DIM", 3)
DIM = par.parval_from_str("grid::DIM")
# Set reference metric related quantities
rfm.reference_metric()
# Register gridfunctions that are needed as input.
# Declare the rank-1 indexed expressions E^{i}, A^{i},
# and \partial^{i} \psi, that are to be evolved in time.
# Derivative variables like these must have an underscore
# in them, so the finite difference module can parse
# the variable name properly.
# e^i
eU = ixp.register_gridfunctions_for_single_rank1("EVOL", "eU")
# \partial_k ( E^i ) --> rank two tensor
eU_dD = ixp.declarerank2("eU_dD", "nosym")
# a^i
aU = ixp.register_gridfunctions_for_single_rank1("EVOL", "aU")
# \partial_k ( a^i ) --> rank two tensor
aU_dD = ixp.declarerank2("aU_dD", "nosym")
# \partial_k partial_m ( a^i ) --> rank three tensor
aU_dDD = ixp.declarerank3("aU_dDD", "sym12")
# \psi is a scalar function that is time evolved
# psi is unused here
_psi = gri.register_gridfunctions("EVOL", ["psi"])
# \Gamma is a scalar function that is time evolved
Gamma = gri.register_gridfunctions("EVOL", ["Gamma"])
# \partial_i \psi
psi_dD = ixp.declarerank1("psi_dD")
# \partial_i \Gamma
Gamma_dD = ixp.declarerank1("Gamma_dD")
# partial_i \partial_j \psi
psi_dDD = ixp.declarerank2("psi_dDD", "sym01")
ghatUU = rfm.ghatUU
GammahatUDD = rfm.GammahatUDD
GammahatUDDdD = rfm.GammahatUDDdD
ReU = rfm.ReU
ReUdD = rfm.ReUdD
ReUdDD = rfm.ReUdDD
# \partial_t a^i = -e^i - \frac{\hat{g}^{ij}\partial_j \varphi}{\text{ReU}[i]}
arhsU = ixp.zerorank1()
for i in range(DIM):
arhsU[i] -= eU[i]
for j in range(DIM):
arhsU[i] -= (ghatUU[i][j] * psi_dD[j]) / ReU[i]
# A^i
AU = ixp.zerorank1()
# \partial_k ( A^i ) --> rank two tensor
AU_dD = ixp.zerorank2()
# \partial_k partial_m ( A^i ) --> rank three tensor
AU_dDD = ixp.zerorank3()
for i in range(DIM):
AU[i] = aU[i] * ReU[i]
for j in range(DIM):
AU_dD[i][j] = aU_dD[i][j] * ReU[i] + aU[i] * ReUdD[i][j]
for k in range(DIM):
AU_dDD[i][j][k] = aU_dDD[i][j][k]*ReU[i] + aU_dD[i][j]*ReUdD[i][k] +\
aU_dD[i][k]*ReUdD[i][j] + aU[i]*ReUdDD[i][j][k]
# Term 1 = \hat{g}^{ij}\partial_j \Gamma
Term1U = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
Term1U[i] += ghatUU[i][j] * Gamma_dD[j]
# Term 2: A^i_{,kj}
Term2UDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Term2UDD[i][j][k] += AU_dDD[i][k][j]
# Term 3: \hat{\Gamma}^i_{mk,j} A^m + \hat{\Gamma}^i_{mk} A^m_{,j}
# + \hat{\Gamma}^i_{dj}A^d_{,k} - \hat{\Gamma}^d_{kj} A^i_{,d}
Term3UDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
Term3UDD[i][j][k] += GammahatUDDdD[i][m][k][j]*AU[m] \
+ GammahatUDD[i][m][k]*AU_dD[m][j] \
+ GammahatUDD[i][m][j]*AU_dD[m][k] \
- GammahatUDD[m][k][j]*AU_dD[i][m]
# Term 4: \hat{\Gamma}^i_{dj}\hat{\Gamma}^d_{mk} A^m -
# \hat{\Gamma}^d_{kj} \hat{\Gamma}^i_{md} A^m
Term4UDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
for d in range(DIM):
Term4UDD[i][j][k] += ( GammahatUDD[i][d][j]*GammahatUDD[d][m][k] \
-GammahatUDD[d][k][j]*GammahatUDD[i][m][d])*AU[m]
# \partial_t E^i = \hat{g}^{ij}\partial_j \Gamma - \hat{\gamma}^{jk}*
# (A^i_{,kj}
# + \hat{\Gamma}^i_{mk,j} A^m + \hat{\Gamma}^i_{mk} A^m_{,j}
# + \hat{\Gamma}^i_{dj} A^d_{,k} - \hat{\Gamma}^d_{kj} A^i_{,d}
# + \hat{\Gamma}^i_{dj}\hat{\Gamma}^d_{mk} A^m
# - \hat{\Gamma}^d_{kj} \hat{\Gamma}^i_{md} A^m)
ErhsU = ixp.zerorank1()
for i in range(DIM):
ErhsU[i] += Term1U[i]
for j in range(DIM):
for k in range(DIM):
ErhsU[i] -= ghatUU[j][k] * (
Term2UDD[i][j][k] + Term3UDD[i][j][k] + Term4UDD[i][j][k])
erhsU = ixp.zerorank1()
for i in range(DIM):
erhsU[i] = ErhsU[i] / ReU[i]
# \partial_t \Gamma = -\hat{g}^{ij} (\partial_i \partial_j \varphi -
# \hat{\Gamma}^k_{ji} \partial_k \varphi)
Gamma_rhs = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
Gamma_rhs -= ghatUU[i][j] * psi_dDD[i][j]
for k in range(DIM):
Gamma_rhs += ghatUU[i][j] * GammahatUDD[k][j][i] * psi_dD[k]
# \partial_t \varphi = -\Gamma
psi_rhs = -Gamma
# \mathcal{G} \equiv \Gamma - \partial_i A^i + \hat{\Gamma}^i_{ji} A^j
G = Gamma
for i in range(DIM):
G -= AU_dD[i][i]
for j in range(DIM):
G += GammahatUDD[i][j][i] * AU[j]
# E^i
EU = ixp.zerorank1()
EU_dD = ixp.zerorank2()
for i in range(DIM):
EU[i] = eU[i] * ReU[i]
for j in range(DIM):
EU_dD[i][j] = eU_dD[i][j] * ReU[i] + eU[i] * ReUdD[i][j]
C = sp.sympify(0)
for i in range(DIM):
C += EU_dD[i][i]
for j in range(DIM):
C += GammahatUDD[i][j][i] * EU[j]
def Convert_to_Cartesian_basis(VU):
# Coordinate transformation from original basis to Cartesian
rfm.reference_metric()
VU_Cart = ixp.zerorank1()
Jac_dxCartU_dxOrigD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dxCartU_dxOrigD[i][j] = sp.diff(rfm.xxCart[i], rfm.xx[j])
for i in range(DIM):
for j in range(DIM):
VU_Cart[i] += Jac_dxCartU_dxOrigD[i][j] * VU[j]
return VU_Cart
AU_Cart = Convert_to_Cartesian_basis(AU)
EU_Cart = Convert_to_Cartesian_basis(EU)
