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 Ricci__generate_symbolic_expressions(): ###################################### # START: GENERATE SYMBOLIC EXPRESSIONS starttime = print_msg_with_timing("3-Ricci tensor", msg="Symbolic", startstop="start") # Evaluate 3-Ricci tensor: import BSSN.BSSN_quantities as Bq par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic", "False") # Register all BSSN gridfunctions if not registered already Bq.BSSN_basic_tensors() # Next compute Ricci tensor Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU() # END: GENERATE SYMBOLIC EXPRESSIONS ###################################### # Must register RbarDD as gridfunctions, as we're outputting them to gridfunctions here: foundit = False for i in range(len(gri.glb_gridfcs_list)): if "RbarDD00" in gri.glb_gridfcs_list[i].name: foundit = True if not foundit: ixp.register_gridfunctions_for_single_rank2("AUXEVOL", "RbarDD", "sym01") 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]) ] print_msg_with_timing("3-Ricci tensor", msg="Symbolic", startstop="stop", starttime=starttime) return Ricci_SymbExpressions
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_RHSs_Ricci__generate_symbolic_expressions(): ###################################### # START: GENERATE SYMBOLIC EXPRESSIONS # Store original finite-differencing order: FD_order_orig = par.parval_from_str("finite_difference::FD_CENTDERIVS_ORDER") # Set new finite-differencing order: par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", FD_order) print("Generating symbolic expressions for BSSN RHSs and 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). cmd.mkdir(os.path.join(outdir,"rfm_files/")) 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 # Next compute Ricci tensor par.set_parval_from_str("BSSN.BSSN_quantities::LeaveRicciSymbolic","False") 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("Finished BSSN symbolic expressions in "+str(end-start)+" seconds.") # Restore original finite-differencing order: par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", FD_order) # 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]) ] 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 [betaU,BSSN_RHSs_SymbExpressions,Ricci_SymbExpressions]
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 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 })