def test_example_BSSN():
import NRPy_param_funcs as par, reference_metric as rfm, BSSN.BSSN_RHSs as Brhs
Parser.clear_namespace()
parse(r"""
% define deltaDD (3D), sym01 hDD (3D), sym01 aDD (3D), vetU (3D)
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign symbolic <H> gammahatDD
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign numeric hDD, aDD, vetU, gammabarDD
% assign metric gammahatDD, gammabarDD
% srepl "\bar{A}" -> "a", "\beta" -> "\text{vet}"
% parse \bar{A}^i_j = \bar{\gamma}^{ik} \bar{A}_{kj}
\begin{align}
% define basis [x, y, z]
%% parse \partial_k \bar{\gamma}_{ij} = \vphantom{upwind} \partial_k h_{ij} + \partial_k \hat{\gamma}_{ij}
% srepl "\text{vet}^<1> \partial_<1>" -> "\text{vet}^<1> \vphantom{upwind} \partial_<1>" %% (inside Lie derivative expansion)
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\text{vet}^k \vphantom{symbolic} \partial_k \text{gammabardet}}{2 \text{gammabardet}})"
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{ij} &= \mathcal{L}_\beta \bar{\gamma}_{ij}
+ \frac{2}{3} \bar{\gamma}_{ij} \left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{ij}
% srepl "K" -> "\text{trK}", "\phi" -> "\text{cf}", "\partial_t \text{cf}" -> "\text{cf_rhs}"
\partial_t \phi &= \mathcal{L}_\beta \phi - \frac{1}{3} \phi \left(\bar{D}_k \beta^k - \alpha K \right)
\end{align}
""")
par.set_parval_from_str('reference_metric::CoordSystem', 'Cartesian')
rfm.reference_metric()
Brhs.BSSN_RHSs()
assert_equal({
'h_rhsDD': h_rhsDD,
'cf_rhs': cf_rhs
}, {
'h_rhsDD': Brhs.h_rhsDD,
'cf_rhs': Brhs.cf_rhs
})
def Ricci__generate_symbolic_expressions():
######################################
# START: GENERATE SYMBOLIC EXPRESSIONS
print("Generating symbolic expressions for Ricci tensor...")
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
# Next compute Ricci tensor
# par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic","False")
RbarDD_already_registered = False
for i in range(len(gri.glb_gridfcs_list)):
if "RbarDD00" in gri.glb_gridfcs_list[i].name:
RbarDD_already_registered = True
if not RbarDD_already_registered:
# We ignore the return value of ixp.register_gridfunctions_for_single_rank2() below
# as it is unused.
ixp.register_gridfunctions_for_single_rank2("AUXEVOL","RbarDD","sym01")
rhs.BSSN_RHSs()
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
# 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()
end = time.time()
print("(BENCH) Finished Ricci symbolic expressions in "+str(end-start)+" seconds.")
# END: GENERATE SYMBOLIC EXPRESSIONS
######################################
Ricci_SymbExpressions = [lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD00"),rhs=Bq.RbarDD[0][0]),
lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD01"),rhs=Bq.RbarDD[0][1]),
lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD02"),rhs=Bq.RbarDD[0][2]),
lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD11"),rhs=Bq.RbarDD[1][1]),
lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD12"),rhs=Bq.RbarDD[1][2]),
lhrh(lhs=gri.gfaccess("auxevol_gfs","RbarDD22"),rhs=Bq.RbarDD[2][2])]
return [Ricci_SymbExpressions]
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 test_example_BSSN():
parse_latex(r"""
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% upwind pattern inside Lie derivative expansion
% srepl -persist "\text{vet}^{<1>} \partial_{<1>}" -> "\text{vet}^{<1>} \vphantom{dupD} \partial_{<1>}"
%% substitute tensor identity (see appropriate BSSN notebook)
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{ij} &= \mathcal{L}_\beta \bar{\gamma}_{ij}
+ \frac{2}{3} \bar{\gamma}_{ij} \left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{ij} \\
% srepl "K" -> "\text{trK}"
% vardef -diff_type=dD 'cf', 'trK'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4\phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
\partial_t \phi &= \mathcal{L}_\beta \phi + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% vardef -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \mathcal{L}_\beta K + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{ij} \bar{A}^{ij}
- e^{-4\phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi\right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% assign -metric='gammabar' 'DeltaUDD'
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \mathcal{L}_\beta \bar{\Lambda}^i + \bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i
+ \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad- 2 \bar{A}^{ij} \left(\partial_j \alpha - 6 \alpha \partial_j \phi\right)
+ 2 \alpha \bar{A}^{jk} \Delta^i_{jk} - \frac{4}{3} \alpha \bar{\gamma}^{ij} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{ij} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ 2 \bar{D}_i \alpha \bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi
- \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{ij} \\
\hat{X}_{ij} &= X_{ij} - \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} X_{kl} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{ij} &= \mathcal{L}_\beta \bar{A}_{ij} - \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} \bar{A}^k_j + \alpha \bar{A}_{ij} K + e^{-4\phi} \hat{X}_{ij} \\
% srepl "\partial_t \alpha" -> "\text{alpha_rhs}"
\partial_t \alpha &= \mathcal{L}_\beta \alpha - 2 \alpha K \\
% srepl "B" -> "\text{bet}"
% vardef -diff_type=dD 'betU'
% srepl "\partial_t \text{vet}" -> "\text{vet_rhs}"
\partial_t \beta^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j \beta^i\right] + B^i \\
% vardef -const 'eta'
% srepl "\partial_t \text{bet}" -> "\text{bet_rhs}"
\partial_t B^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j B^i\right]
+ \frac{3}{4} \left(\partial_t \bar{\Lambda}^i - \left[\beta^j \vphantom{dupD} \bar{D}_j \bar{\Lambda}^i\right]\right) - \eta B^i \\
% parse \bar{R} = \bar{\gamma}^{ij} \bar{R}_{ij}
% srepl "\bar{D}^2" -> "\bar{D}^i \bar{D}_i", "\mathcal{<1>}" -> "<1>"
\mathcal{H} &= \frac{2}{3} K^2 - \bar{A}_{ij} \bar{A}^{ij}
+ e^{-4\phi} \left(\bar{R} - 8 \bar{D}^i \phi \bar{D}_i \phi - 8 \bar{D}^2 \phi\right) \\
\mathcal{M}^i &= e^{-4\phi} \left(\bar{D}_j \bar{A}^{ij} + 6 \bar{A}^{ij} \partial_j \phi
- \frac{2}{3} \bar{\gamma}^{ij} \partial_j K\right) \\
\bar{R}_{ij} &= -\frac{1}{2} \bar{\gamma}^{kl} \hat{D}_k \hat{D}_l \bar{\gamma}_{ij}
+ \frac{1}{2} \left(\bar{\gamma}_{ki} \hat{D}_j \bar{\Lambda}^k + \bar{\gamma}_{kj} \hat{D}_i \bar{\Lambda}^k\right)
+ \frac{1}{2} \Delta^k \left(\Delta_{ijk} + \Delta_{jik}\right) \\%
&\qquad+ \bar{\gamma}^{kl} \left(\Delta^m_{ki} \Delta_{jml} + \Delta^m_{kj} \Delta_{iml} + \Delta^m_{ik} \Delta_{mjl}\right)
\end{align}
""",
ignore_warning=True)
par.set_parval_from_str('reference_metric::CoordSystem', 'Cartesian')
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic',
'True')
rfm.reference_metric()
Brhs.BSSN_RHSs()
gaugerhs.BSSN_gauge_RHSs()
bssncon.BSSN_constraints()
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic',
'False')
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
assert_equal(
{
'h_rhsDD': h_rhsDD,
'cf_rhs': cf_rhs,
'trK_rhs': trK_rhs,
'Lambdabar_rhsU': Lambdabar_rhsU,
'a_rhsDD': a_rhsDD,
'alpha_rhs': alpha_rhs,
'vet_rhsU': vet_rhsU,
'bet_rhsU': bet_rhsU,
'H': H,
'MU': MU,
'RbarDD': RbarDD
}, {
'h_rhsDD': Brhs.h_rhsDD,
'cf_rhs': Brhs.cf_rhs,
'trK_rhs': Brhs.trK_rhs,
'Lambdabar_rhsU': Brhs.Lambdabar_rhsU,
'a_rhsDD': Brhs.a_rhsDD,
'alpha_rhs': gaugerhs.alpha_rhs,
'vet_rhsU': gaugerhs.vet_rhsU,
'bet_rhsU': gaugerhs.bet_rhsU,
'H': bssncon.H,
'MU': bssncon.MU,
'RbarDD': Bq.RbarDD
},
suppress_message=True)
def test_example_BSSN():
import NRPy_param_funcs as par, reference_metric as rfm
import BSSN.BSSN_RHSs as Brhs, BSSN.BSSN_quantities as Bq
Parser.clear_namespace()
parse(r"""
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef 'deltaDD', 'vetU', 'lambdaU'
% vardef -numeric -sym01 'hDD', 'aDD', 'RbarDD'
% assign -numeric 'cf', 'alpha', 'trK', 'vetU', 'lambdaU'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -symbolic <H> -metric 'gammahatDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -numeric -metric 'gammabarDD'
\begin{align}
%% replace LaTeX variable(s) with BSSN variable(s)
% srepl "\bar{A}" -> "\text{a}", "\beta" -> "\text{vet}", "K" -> "\text{trK}", "\bar{\Lambda}" -> "\text{lambda}"
% srepl "e^{{-4\phi}}" -> "\text{cf}^{{2}}"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -global "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl -global "\partial_<1> \text{phi}" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl -global "\partial_<1> (\text{phi})" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
%% upwind pattern inside Lie derivative expansion
% srepl -global "\text{vet}^<1> \partial_<1>" -> "\text{vet}^<1> \vphantom{upwind} \partial_<1>"
%% enforce metric constraint gammabardet == gammahatdet
% srepl -global "\bar{D}^i \bar{D}_k \text{vet}^k" -> "(\bar{D}^i \partial_k \text{vet}^k)"
% srepl -global "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% parse \bar{A}^i_j = \bar{\gamma}^{ik} \bar{A}_{kj}
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{ij} &= \mathcal{L}_\beta \bar{\gamma}_{ij}
+ \frac{2}{3} \bar{\gamma}_{ij} \left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right)
- 2 \alpha \bar{A}_{ij} \\
\partial_t \phi &= \mathcal{L}_\beta \phi
+ \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% parse \bar{A}^{ij} = \bar{\gamma}^{ik} \bar{\gamma}^{jl} \bar{A}_{kl}
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \mathcal{L}_\beta K
+ \frac{1}{3} \alpha K^{{2}}
+ \alpha \bar{A}_{ij} \bar{A}^{ij}
- e^{{-4\phi}} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi\right) \\
% parse \Pi^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Pi_{ijk} = \bar{\gamma}_{il} \Pi^l_{jk}
% parse \Pi^k = \bar{\gamma}^{ij} \Pi^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \mathcal{L}_\beta \bar{\Lambda}^i + \bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i
+ \frac{2}{3} \Pi^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad- 2 \bar{A}^{ij} (\partial_j \alpha - 6 \alpha \partial_j \phi)
+ 2 \alpha \bar{A}^{jk} \Pi^i_{jk} - \frac{4}{3} \alpha \bar{\gamma}^{ij} \partial_j K \\
X_{ij} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ 2 \bar{D}_i \alpha \bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi
- \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{ij} \\
\hat{X}_{ij} &= X_{ij} - \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} X_{kl} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{ij} &= \mathcal{L}_\beta \bar{A}_{ij}
- \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} \bar{A}^k_j
+ \alpha \bar{A}_{ij} K
+ e^{{-4\phi}} \hat{X}_{ij} \\
\bar{R}_{ij} &= -\frac{1}{2} \bar{\gamma}^{kl} \hat{D}_k \hat{D}_l \bar{\gamma}_{ij}
+ \frac{1}{2} (\bar{\gamma}_{ki} \hat{D}_j \bar{\Lambda}^k + \bar{\gamma}_{kj} \hat{D}_i \bar{\Lambda}^k)
+ \frac{1}{2} \Pi^k (\Pi_{ijk} + \Pi_{jik}) \\%
&\qquad+ \bar{\gamma}^{kl} (\Pi^m_{ki} \Pi_{jml} + \Pi^m_{kj} \Pi_{iml} + \Pi^m_{ik} \Pi_{mjl})
\end{align}
""")
par.set_parval_from_str('reference_metric::CoordSystem', 'Cartesian')
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic',
'True')
rfm.reference_metric()
Brhs.BSSN_RHSs()
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic',
'False')
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
assert_equal(
{
'h_rhsDD': h_rhsDD,
'cf_rhs': cf_rhs,
'trK_rhs': trK_rhs,
'Lambdabar_rhsU': Lambdabar_rhsU,
'a_rhsDD': a_rhsDD,
'RbarDD': RbarDD
}, {
'h_rhsDD': Brhs.h_rhsDD,
'cf_rhs': Brhs.cf_rhs,
'trK_rhs': Brhs.trK_rhs,
'Lambdabar_rhsU': Brhs.Lambdabar_rhsU,
'a_rhsDD': Brhs.a_rhsDD,
'RbarDD': Bq.RbarDD
})
def BSSN_RHSs__generate_symbolic_expressions(
LapseCondition="OnePlusLog",
ShiftCondition="GammaDriving2ndOrder_Covariant",
enable_KreissOliger_dissipation=True,
enable_stress_energy_source_terms=False,
leave_Ricci_symbolic=True):
######################################
# START: GENERATE SYMBOLIC EXPRESSIONS
starttime = print_msg_with_timing("BSSN_RHSs",
msg="Symbolic",
startstop="start")
# Returns None if enable_stress_energy_source_terms==False; otherwise returns symb expressions for T4UU
T4UU = register_stress_energy_source_terms_return_T4UU(
enable_stress_energy_source_terms)
# Evaluate BSSN RHSs:
import BSSN.BSSN_quantities as Bq
par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic",
str(leave_Ricci_symbolic))
rhs.BSSN_RHSs()
if enable_stress_energy_source_terms:
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]
par.set_parval_from_str("BSSN.BSSN_gauge_RHSs::LapseEvolutionOption",
LapseCondition)
par.set_parval_from_str("BSSN.BSSN_gauge_RHSs::ShiftEvolutionOption",
ShiftCondition)
gaugerhs.BSSN_gauge_RHSs() # Can depend on above RHSs
# Restore BSSN.BSSN_quantities::LeaveRicciSymbolic to False
par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic",
"False")
# Add Kreiss-Oliger dissipation to the BSSN RHSs:
if enable_KreissOliger_dissipation:
thismodule = "KO_Dissipation"
diss_strength = par.Cparameters(
"REAL", thismodule, "diss_strength", 0.1
) # *Bq.cf # *Bq.cf*Bq.cf*Bq.cf # cf**1 is found better than cf**4 over the long term.
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
# END: GENERATE SYMBOLIC EXPRESSIONS
######################################
lhs_names = ["alpha", "cf", "trK"]
rhs_exprs = [gaugerhs.alpha_rhs, rhs.cf_rhs, rhs.trK_rhs]
for i in range(3):
lhs_names.append("betU" + str(i))
rhs_exprs.append(gaugerhs.bet_rhsU[i])
lhs_names.append("lambdaU" + str(i))
rhs_exprs.append(rhs.lambda_rhsU[i])
lhs_names.append("vetU" + str(i))
rhs_exprs.append(gaugerhs.vet_rhsU[i])
for j in range(i, 3):
lhs_names.append("aDD" + str(i) + str(j))
rhs_exprs.append(rhs.a_rhsDD[i][j])
lhs_names.append("hDD" + str(i) + str(j))
rhs_exprs.append(rhs.h_rhsDD[i][j])
# Sort the lhss list alphabetically, and rhss to match.
# This ensures the RHSs are evaluated in the same order
# they're allocated in memory:
lhs_names, rhs_exprs = [
list(x) for x in zip(
*sorted(zip(lhs_names, rhs_exprs), key=lambda pair: pair[0]))
]
# Declare the list of lhrh's
BSSN_RHSs_SymbExpressions = []
for var in range(len(lhs_names)):
BSSN_RHSs_SymbExpressions.append(
lhrh(lhs=gri.gfaccess("rhs_gfs", lhs_names[var]),
rhs=rhs_exprs[var]))
print_msg_with_timing("BSSN_RHSs",
msg="Symbolic",
startstop="stop",
starttime=starttime)
return [betaU, BSSN_RHSs_SymbExpressions]
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]
# Step 2.b: Implement the harmonic slicing lapse condition
elif par.parval_from_str(thismodule +
"::LapseEvolutionOption") == "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)"
)
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 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 par.parval_from_str(
thismodule +
"::ShiftEvolutionOption") == "GammaDriving2ndOrder_Covariant":
# 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()
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 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] * 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]