def BSSN_gauge_RHSs():
# 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: 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.f: Define needed BSSN quantities:
# Declare scalars & tensors (in terms of rescaled BSSN quantities)
Bq.BSSN_basic_tensors()
Bq.betaU_derivs()
# Declare BSSN_RHSs (excluding the time evolution equations for the gauge conditions),
# if they haven't already been declared.
if Brhs.have_already_called_BSSN_RHSs_function == False:
print(
"BSSN_gauge_RHSs() Error: You must call BSSN_RHSs() before calling BSSN_gauge_RHSs()."
)
sys.exit(1)
# Step 2: Lapse conditions
LapseEvolOption = par.parval_from_str(thismodule +
"::LapseEvolutionOption")
# Step 2.a: The 1+log lapse condition:
# \partial_t \alpha = \beta^i \alpha_{,i} - 2*\alpha*K
# First import expressions from BSSN_quantities
cf = Bq.cf
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
# Implement the 1+log lapse condition
global alpha_rhs
alpha_rhs = sp.sympify(0)
if LapseEvolOption == "OnePlusLog":
alpha_rhs = -2 * alpha * trK
alpha_dupD = ixp.declarerank1("alpha_dupD")
for i in range(DIM):
alpha_rhs += betaU[i] * alpha_dupD[i]
# Step 2.b: Implement the harmonic slicing lapse condition
elif LapseEvolOption == "HarmonicSlicing":
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "W":
alpha_rhs = -3 * cf**(-4) * Brhs.cf_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "phi":
alpha_rhs = 6 * sp.exp(6 * cf) * Brhs.cf_rhs
else:
print(
"Error LapseEvolutionOption==HarmonicSlicing unsupported for EvolvedConformalFactor_cf!=(W or phi)"
)
sys.exit(1)
# Step 2.c: Frozen lapse
# \partial_t \alpha = 0
elif LapseEvolOption == "Frozen":
alpha_rhs = sp.sympify(0)
else:
print("Error: " + thismodule + "::LapseEvolutionOption == " +
LapseEvolOption + " not supported!")
sys.exit(1)
# Step 3.a: Set \partial_t \beta^i
# First check that ShiftEvolutionOption parameter choice is supported.
ShiftEvolOption = par.parval_from_str(thismodule +
"::ShiftEvolutionOption")
if ShiftEvolOption != "Frozen" and \
ShiftEvolOption != "GammaDriving2ndOrder_NoCovariant" and \
ShiftEvolOption != "GammaDriving2ndOrder_Covariant" and \
ShiftEvolOption != "GammaDriving2ndOrder_Covariant__Hatted" and \
ShiftEvolOption != "GammaDriving1stOrder_Covariant" and \
ShiftEvolOption != "GammaDriving1stOrder_Covariant__Hatted":
print("Error: ShiftEvolutionOption == " + ShiftEvolOption +
" unsupported!")
sys.exit(1)
# Next import expressions from BSSN_quantities
BU = Bq.BU
betU = Bq.betU
betaU_dupD = Bq.betaU_dupD
# Define needed quantities
beta_rhsU = ixp.zerorank1()
B_rhsU = ixp.zerorank1()
# In the case of Frozen shift condition, we
# explicitly set the betaU and BU RHS's to zero
# instead of relying on the ixp.zerorank1()'s above,
# for safety.
if ShiftEvolOption == "Frozen":
for i in range(DIM):
beta_rhsU[i] = sp.sympify(0)
BU[i] = sp.sympify(0)
if ShiftEvolOption == "GammaDriving2ndOrder_NoCovariant":
# Step 3.a.i: Compute right-hand side of beta^i
# * \partial_t \beta^i = \beta^j \beta^i_{,j} + B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]
for j in range(DIM):
beta_rhsU[i] += betaU[j] * betaU_dupD[i][j]
# Compute right-hand side of B^i:
eta = par.Cparameters("REAL", thismodule, ["eta"], 2.0)
# Step 3.a.ii: Compute right-hand side of B^i
# * \partial_t B^i = \beta^j B^i_{,j} + 3/4 * \partial_0 \Lambda^i - eta B^i
# Step 3.a.iii: Define BU_dupD, in terms of derivative of rescaled variable \bet^i
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j] * rfm.ReU[i] + betU[
i] * rfm.ReUdD[i][j]
# Step 3.a.iv: Compute \partial_0 \bar{\Lambda}^i = (\partial_t - \beta^i \partial_i) \bar{\Lambda}^j
Lambdabar_partial0 = ixp.zerorank1()
for i in range(DIM):
Lambdabar_partial0[i] = Brhs.Lambdabar_rhsU[i]
for i in range(DIM):
for j in range(DIM):
Lambdabar_partial0[j] += -betaU[i] * Brhs.LambdabarU_dupD[j][i]
# Step 3.a.v: Evaluate RHS of B^i:
for i in range(DIM):
B_rhsU[i] += sp.Rational(3,
4) * Lambdabar_partial0[i] - eta * BU[i]
for j in range(DIM):
B_rhsU[i] += betaU[j] * BU_dupD[i][j]
# Step 3.b: The right-hand side of the \partial_t \beta^i equation
if "GammaDriving2ndOrder_Covariant" in ShiftEvolOption:
# Step 3.b Option 2: \partial_t \beta^i = \left[\beta^j \bar{D}_j \beta^i\right] + B^{i}
# First we need GammabarUDD, defined in Bq.gammabar__inverse_and_derivs()
Bq.gammabar__inverse_and_derivs()
ConnectionUDD = Bq.GammabarUDD
# If instead we wish to use the Hatted covariant derivative, we replace
# ConnectionUDD with GammahatUDD:
if ShiftEvolOption == "GammaDriving2ndOrder_Covariant__Hatted":
ConnectionUDD = rfm.GammahatUDD
# Then compute right-hand side:
# Term 1: \beta^j \beta^i_{,j}
for i in range(DIM):
for j in range(DIM):
beta_rhsU[i] += betaU[j] * betaU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} \beta^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
beta_rhsU[
i] += betaU[j] * ConnectionUDD[i][m][j] * betaU[m]
# Term 3: B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]
if "GammaDriving2ndOrder_Covariant" in ShiftEvolOption:
ConnectionUDD = Bq.GammabarUDD
# If instead we wish to use the Hatted covariant derivative, we replace
# ConnectionUDD with GammahatUDD:
if ShiftEvolOption == "GammaDriving2ndOrder_Covariant__Hatted":
ConnectionUDD = rfm.GammahatUDD
# Step 3.c: Covariant option:
# \partial_t B^i = \beta^j \bar{D}_j B^i
# + \frac{3}{4} ( \partial_t \bar{\Lambda}^{i} - \beta^j \bar{D}_j \bar{\Lambda}^{i} )
# - \eta B^{i}
# = \beta^j B^i_{,j} + \beta^j \bar{\Gamma}^i_{mj} B^m
# + \frac{3}{4}[ \partial_t \bar{\Lambda}^{i}
# - \beta^j (\bar{\Lambda}^i_{,j} + \bar{\Gamma}^i_{mj} \bar{\Lambda}^m)]
# - \eta B^{i}
# Term 1, part a: First compute B^i_{,j} using upwinded derivative
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j] * rfm.ReU[i] + betU[
i] * rfm.ReUdD[i][j]
# Term 1: \beta^j B^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += betaU[j] * BU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} B^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += betaU[j] * ConnectionUDD[i][m][j] * BU[m]
# Term 3: \frac{3}{4}\partial_t \bar{\Lambda}^{i}
for i in range(DIM):
B_rhsU[i] += sp.Rational(3, 4) * Brhs.Lambdabar_rhsU[i]
# Term 4: -\frac{3}{4}\beta^j \bar{\Lambda}^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += -sp.Rational(
3, 4) * betaU[j] * Brhs.LambdabarU_dupD[i][j]
# Term 5: -\frac{3}{4}\beta^j \bar{\Gamma}^i_{mj} \bar{\Lambda}^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += -sp.Rational(3, 4) * betaU[j] * ConnectionUDD[
i][m][j] * Bq.LambdabarU[m]
# Term 6: - \eta B^i
# eta is a free parameter; we declare it here:
eta = par.Cparameters("REAL", thismodule, ["eta"], 2.0)
for i in range(DIM):
B_rhsU[i] += -eta * BU[i]
if "GammaDriving1stOrder_Covariant" in ShiftEvolOption:
# Step 3.c: \partial_t \beta^i = \left[\beta^j \bar{D}_j \beta^i\right] + 3/4 Lambdabar^i - eta*beta^i
# First set \partial_t B^i = 0:
B_rhsU = ixp.zerorank1() # \partial_t B^i = 0
# Second, set \partial_t beta^i RHS:
# Compute covariant advection term:
# We need GammabarUDD, defined in Bq.gammabar__inverse_and_derivs()
Bq.gammabar__inverse_and_derivs()
ConnectionUDD = Bq.GammabarUDD
# If instead we wish to use the Hatted covariant derivative, we replace
# ConnectionUDD with GammahatUDD:
if ShiftEvolOption == "GammaDriving1stOrder_Covariant__Hatted":
ConnectionUDD = rfm.GammahatUDD
# Term 1: \beta^j \beta^i_{,j}
for i in range(DIM):
for j in range(DIM):
beta_rhsU[i] += betaU[j] * betaU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} \beta^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
beta_rhsU[
i] += betaU[j] * ConnectionUDD[i][m][j] * betaU[m]
# Term 3: 3/4 Lambdabar^i - eta*beta^i
eta = par.Cparameters("REAL", thismodule, ["eta"], 2.0)
for i in range(DIM):
beta_rhsU[i] += sp.Rational(3,
4) * Bq.LambdabarU[i] - eta * betaU[i]
# Step 4: Rescale the BSSN gauge RHS quantities so that the evolved
# variables may remain smooth across coord singularities
global vet_rhsU, bet_rhsU
vet_rhsU = ixp.zerorank1()
bet_rhsU = ixp.zerorank1()
for i in range(DIM):
vet_rhsU[i] = beta_rhsU[i] / rfm.ReU[i]
bet_rhsU[i] = B_rhsU[i] / rfm.ReU[i]
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 BSSN_gauge_RHSs():
# 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: 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.f: Define needed BSSN quantities:
# Declare scalars & tensors (in terms of rescaled BSSN quantities)
Bq.BSSN_basic_tensors()
Bq.betaU_derivs()
# Declare BSSN_RHSs (excluding the time evolution equations for the gauge conditions)
Brhs.BSSN_RHSs()
# Step 2.a: The 1+log lapse condition:
# \partial_t \alpha = \beta^i \alpha_{,i} - 2*\alpha*K
# First import expressions from BSSN_quantities
cf = Bq.cf
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
# Implement the 1+log lapse condition
global alpha_rhs
alpha_rhs = sp.sympify(0)
if par.parval_from_str(thismodule +
"::LapseEvolutionOption") == "OnePlusLog":
alpha_rhs = -2 * alpha * trK
alpha_dupD = ixp.declarerank1("alpha_dupD")
for i in range(DIM):
alpha_rhs += betaU[i] * alpha_dupD[i]
# Implement the harmonic slicing lapse condition
elif par.parval_from_str(thismodule +
"::LapseEvolutionOption") == "HarmonicSlicing":
if par.parval_from_str("BSSN.BSSN_quantities::ConformalFactor") == "W":
alpha_rhs = -3 * cf**(-4) * Brhs.cf_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::ConformalFactor") == "phi":
alpha_rhs = 6 * sp.exp(6 * cf) * Brhs.cf_rhs
else:
print(
"Error LapseEvolutionOption==HarmonicSlicing unsupported for ConformalFactor!=(W or phi)"
)
exit(1)
# Step 2.c: Frozen lapse
# \partial_t \alpha = 0
elif par.parval_from_str(thismodule +
"::LapseEvolutionOption") == "Frozen":
alpha_rhs = sp.sympify(0)
else:
print("Error: " + thismodule + "::LapseEvolutionOption == " +
par.parval_from_str(thismodule + "::LapseEvolutionOption") +
" not supported!")
exit(1)
# Step 3.a: Set \partial_t \beta^i
# First import expressions from BSSN_quantities
BU = Bq.BU
betU = Bq.betU
betaU_dupD = Bq.betaU_dupD
# Define needed quantities
beta_rhsU = ixp.zerorank1()
B_rhsU = ixp.zerorank1()
if par.parval_from_str(
thismodule +
"::ShiftEvolutionOption") == "GammaDriving2ndOrder_NoCovariant":
# Step 3.a.i: Compute right-hand side of beta^i
# * \partial_t \beta^i = \beta^j \beta^i_{,j} + B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]
for j in range(DIM):
beta_rhsU[i] += betaU[j] * betaU_dupD[i][j]
# Compute right-hand side of B^i:
eta = par.Cparameters("REAL", thismodule, ["eta"])
# Step 3.a.ii: Compute right-hand side of B^i
# * \partial_t B^i = \beta^j B^i_{,j} + 3/4 * \partial_0 \Lambda^i - eta B^i
# Step 15b: Define BU_dupD, in terms of derivative of rescaled variable \bet^i
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j] * rfm.ReU[i] + betU[
i] * rfm.ReUdD[i][j]
# Step 15c: Compute \partial_0 \bar{\Lambda}^i = (\partial_t - \beta^i \partial_i) \bar{\Lambda}^j
Lambdabar_partial0 = ixp.zerorank1()
for i in range(DIM):
Lambdabar_partial0[i] = Brhs.Lambdabar_rhsU[i]
for i in range(DIM):
for j in range(DIM):
Lambdabar_partial0[j] += -betaU[i] * Brhs.LambdabarU_dupD[j][i]
# Step 15d: Evaluate RHS of B^i:
for i in range(DIM):
B_rhsU[i] += sp.Rational(3,
4) * Lambdabar_partial0[i] - eta * BU[i]
for j in range(DIM):
B_rhsU[i] += betaU[j] * BU_dupD[i][j]
if par.parval_from_str(
thismodule +
"::ShiftEvolutionOption") == "GammaDriving2ndOrder_Covariant":
# Step 14 Option 2: \partial_t \beta^i = \left[\beta^j \bar{D}_j \beta^i\right] + B^{i}
# First we need GammabarUDD, defined in Bq.gammabar__inverse_and_derivs()
Bq.gammabar__inverse_and_derivs()
GammabarUDD = Bq.GammabarUDD
# Then compute right-hand side:
# Term 1: \beta^j \beta^i_{,j}
for i in range(DIM):
for j in range(DIM):
beta_rhsU[i] += betaU[j] * betaU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} \beta^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
beta_rhsU[i] += betaU[j] * GammabarUDD[i][m][j] * betaU[m]
# Term 3: B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]
if par.parval_from_str(
thismodule +
"::ShiftEvolutionOption") == "GammaDriving2ndOrder_Covariant":
# Step 15: Covariant option:
# \partial_t B^i = \beta^j \bar{D}_j B^i
# + \frac{3}{4} ( \partial_t \bar{\Lambda}^{i} - \beta^j \bar{D}_j \bar{\Lambda}^{i} )
# - \eta B^{i}
# = \beta^j B^i_{,j} + \beta^j \bar{\Gamma}^i_{mj} B^m
# + \frac{3}{4}[ \partial_t \bar{\Lambda}^{i}
# - \beta^j (\bar{\Lambda}^i_{,j} + \bar{\Gamma}^i_{mj} \bar{\Lambda}^m)]
# - \eta B^{i}
# Term 1, part a: First compute B^i_{,j} using upwinded derivative
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j] * rfm.ReU[i] + betU[
i] * rfm.ReUdD[i][j]
# Term 1: \beta^j B^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += betaU[j] * BU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} B^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += betaU[j] * GammabarUDD[i][m][j] * BU[m]
# Term 3: \frac{3}{4}\partial_t \bar{\Lambda}^{i}
for i in range(DIM):
B_rhsU[i] += sp.Rational(3, 4) * Brhs.Lambdabar_rhsU[i]
# Term 4: -\frac{3}{4}\beta^j \bar{\Lambda}^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += -sp.Rational(
3, 4) * betaU[j] * Brhs.LambdabarU_dupD[i][j]
# Term 5: -\frac{3}{4}\beta^j \bar{\Gamma}^i_{mj} \bar{\Lambda}^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += -sp.Rational(3, 4) * betaU[j] * GammabarUDD[
i][m][j] * Bq.LambdabarU[m]
# Term 6: - \eta B^i
# eta is a free parameter; we declare it here:
eta = par.Cparameters("REAL", thismodule, ["eta"])
for i in range(DIM):
B_rhsU[i] += -eta * BU[i]
# Step 4: Rescale the BSSN gauge RHS quantities so that the evolved
# variables may remain smooth across coord singularities
global vet_rhsU, bet_rhsU
vet_rhsU = ixp.zerorank1()
bet_rhsU = ixp.zerorank1()
for i in range(DIM):
vet_rhsU[i] = beta_rhsU[i] / rfm.ReU[i]
bet_rhsU[i] = B_rhsU[i] / rfm.ReU[i]
