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_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 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 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 Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(CoordType_in, ADM_input_function_name, Ccodesdir = "BSSN", pointer_to_ID_inputs=False,loopopts=",oldloops"): # The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations. # To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in # the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3. # Step 0: Set spatial dimension (must be 3 for BSSN) DIM = 3 # Step 1: All ADM initial data quantities are now functions of xx0,xx1,xx2, but # they are still in the Spherical or Cartesian basis. We can now directly apply # Jacobian transformations to get them in the correct xx0,xx1,xx2 basis: # All input quantities are in terms of r,th,ph or x,y,z. We want them in terms # of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace # r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2 # as defined for this particular reference metric in reference_metric.py's # xxSph[] or xx_to_Cart[], respectively: # Define the input variables: gammaSphorCartDD = ixp.declarerank2("gammaSphorCartDD", "sym01") KSphorCartDD = ixp.declarerank2("KSphorCartDD", "sym01") alphaSphorCart = sp.symbols("alphaSphorCart") betaSphorCartU = ixp.declarerank1("betaSphorCartU") BSphorCartU = ixp.declarerank1("BSphorCartU") # Make sure that rfm.reference_metric() has been called. # We'll need the variables it defines throughout this module. if rfm.have_already_called_reference_metric_function == False: print("Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without") print(" first setting up reference metric, by calling rfm.reference_metric().") sys.exit(1) r_th_ph_or_Cart_xyz_oID_xx = [] if CoordType_in == "Spherical": r_th_ph_or_Cart_xyz_oID_xx = rfm.xxSph elif CoordType_in == "Cartesian": r_th_ph_or_Cart_xyz_oID_xx = rfm.xx_to_Cart else: print("Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords.") sys.exit(1) # Step 2: All ADM initial data quantities are now functions of xx0,xx1,xx2, but # they are still in the Spherical or Cartesian basis. We can now directly apply # Jacobian transformations to get them in the correct xx0,xx1,xx2 basis: # alpha is a scalar, so no Jacobian transformation is necessary. alpha = alphaSphorCart Jac_dUSphorCart_dDrfmUD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(r_th_ph_or_Cart_xyz_oID_xx[i], rfm.xx[j]) Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(Jac_dUSphorCart_dDrfmUD) betaU = ixp.zerorank1() BU = ixp.zerorank1() gammaDD = ixp.zerorank2() KDD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j] BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j] for k in range(DIM): for l in range(DIM): gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * \ gammaSphorCartDD[k][l] KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l] # Step 3: All ADM quantities were input into this function in the Spherical or Cartesian # basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above, # we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2. # Here we convert ADM quantities in the "rfm" basis to their BSSN Curvilinear # counterparts, for all BSSN quantities *except* lambda^i: import BSSN.BSSN_in_terms_of_ADM as BitoA BitoA.gammabarDD_hDD(gammaDD) BitoA.trK_AbarDD_aDD(gammaDD, KDD) BitoA.cf_from_gammaDD(gammaDD) BitoA.betU_vetU(betaU, BU) hDD = BitoA.hDD trK = BitoA.trK aDD = BitoA.aDD cf = BitoA.cf vetU = BitoA.vetU betU = BitoA.betU # Step 4: Compute $\bar{\Lambda}^i$ (Eqs. 4 and 5 of # [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)), # from finite-difference derivatives of rescaled metric # quantities $h_{ij}$: # \bar{\Lambda}^i = \bar{\gamma}^{jk}\left(\bar{\Gamma}^i_{jk} - \hat{\Gamma}^i_{jk}\right). # The reference_metric.py module provides us with analytic expressions for # $\hat{\Gamma}^i_{jk}$, so here we need only compute # finite-difference expressions for $\bar{\Gamma}^i_{jk}$, based on # the values for $h_{ij}$ provided in the initial data. Once # $\bar{\Lambda}^i$ has been computed, we apply the usual rescaling # procedure: # \lambda^i = \bar{\Lambda}^i/\text{ReU[i]}, # and then output the result to a C file using the NRPy+ # finite-difference C output routine. # We will need all BSSN gridfunctions to be defined, as well as # expressions for gammabarDD_dD in terms of exact derivatives of # the rescaling matrix and finite-difference derivatives of # hDD's. This functionality is provided by BSSN.BSSN_unrescaled_and_barred_vars, # which we call here to overwrite above definitions of gammabarDD,gammabarUU, etc. Bq.gammabar__inverse_and_derivs() # Provides gammabarUU and GammabarUDD gammabarUU = Bq.gammabarUU GammabarUDD = Bq.GammabarUDD # Next evaluate \bar{\Lambda}^i, based on GammabarUDD above and GammahatUDD # (from the reference metric): LambdabarU = ixp.zerorank1() for i in range(DIM): for j in range(DIM): for k in range(DIM): LambdabarU[i] += gammabarUU[j][k] * (GammabarUDD[i][j][k] - rfm.GammahatUDD[i][j][k]) # Finally apply rescaling: # lambda^i = Lambdabar^i/\text{ReU[i]} lambdaU = ixp.zerorank1() for i in range(DIM): lambdaU[i] = LambdabarU[i] / rfm.ReU[i] if ADM_input_function_name == "DoNotOutputADMInputFunction": return hDD,aDD,trK,vetU,betU,alpha,cf,lambdaU # Step 5.A: Output files containing finite-differenced lambdas. outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False" lambdaU_expressions = [lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU0"), rhs=lambdaU[0]), lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU1"), rhs=lambdaU[1]), lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU2"), rhs=lambdaU[2])] desc = "Output lambdaU[i] for BSSN, built using finite-difference derivatives." name = "ID_BSSN_lambdas" params = "const paramstruct *restrict params,REAL *restrict xx[3],REAL *restrict in_gfs" preloop = "" enableCparameters=True if "oldloops" in loopopts: params = "const int Nxx[3],const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],const REAL dxx[3],REAL *in_gfs" enableCparameters=False preloop = """ const REAL invdx0 = 1.0/dxx[0]; const REAL invdx1 = 1.0/dxx[1]; const REAL invdx2 = 1.0/dxx[2]; """ outCfunction( outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params, preloop=preloop, body=fin.FD_outputC("returnstring", lambdaU_expressions, outCparams), loopopts="InteriorPoints,Read_xxs"+loopopts, enableCparameters=enableCparameters) # Step 5: Output all ADM-to-BSSN expressions to a C function. This function # must first call the ID_ADM_SphorCart() defined above. Using these # Spherical or Cartesian data, it sets up all quantities needed for # BSSNCurvilinear initial data, *except* $\lambda^i$, which must be # computed from numerical data using finite-difference derivatives. ID_inputs_param = "ID_inputs other_inputs," if pointer_to_ID_inputs == True: ID_inputs_param = "ID_inputs *other_inputs," desc = "Write BSSN variables in terms of ADM variables at a given point xx0,xx1,xx2" name = "ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs" enableCparameters=True params = "const paramstruct *restrict params, " if "oldloops" in loopopts: enableCparameters=False params = "" params += "const int i0i1i2[3], const REAL xx0xx1xx2[3]," + ID_inputs_param + """ REAL *hDD00,REAL *hDD01,REAL *hDD02,REAL *hDD11,REAL *hDD12,REAL *hDD22, REAL *aDD00,REAL *aDD01,REAL *aDD02,REAL *aDD11,REAL *aDD12,REAL *aDD22, REAL *trK, REAL *vetU0,REAL *vetU1,REAL *vetU2, REAL *betU0,REAL *betU1,REAL *betU2, REAL *alpha, REAL *cf""" outCparams = "preindent=1,outCverbose=False,includebraces=False" outCfunction( outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params, body=""" REAL gammaSphorCartDD00,gammaSphorCartDD01,gammaSphorCartDD02, gammaSphorCartDD11,gammaSphorCartDD12,gammaSphorCartDD22; REAL KSphorCartDD00,KSphorCartDD01,KSphorCartDD02, KSphorCartDD11,KSphorCartDD12,KSphorCartDD22; REAL alphaSphorCart,betaSphorCartU0,betaSphorCartU1,betaSphorCartU2; REAL BSphorCartU0,BSphorCartU1,BSphorCartU2; const REAL xx0 = xx0xx1xx2[0]; const REAL xx1 = xx0xx1xx2[1]; const REAL xx2 = xx0xx1xx2[2]; REAL xyz_or_rthph[3];\n""" + outputC(r_th_ph_or_Cart_xyz_oID_xx[0:3], ["xyz_or_rthph[0]", "xyz_or_rthph[1]", "xyz_or_rthph[2]"], "returnstring", outCparams + ",CSE_enable=False") + " " + ADM_input_function_name + """(params,i0i1i2, xyz_or_rthph, other_inputs, &gammaSphorCartDD00,&gammaSphorCartDD01,&gammaSphorCartDD02, &gammaSphorCartDD11,&gammaSphorCartDD12,&gammaSphorCartDD22, &KSphorCartDD00,&KSphorCartDD01,&KSphorCartDD02, &KSphorCartDD11,&KSphorCartDD12,&KSphorCartDD22, &alphaSphorCart,&betaSphorCartU0,&betaSphorCartU1,&betaSphorCartU2, &BSphorCartU0,&BSphorCartU1,&BSphorCartU2); // Next compute all rescaled BSSN curvilinear quantities:\n""" + outputC([hDD[0][0], hDD[0][1], hDD[0][2], hDD[1][1], hDD[1][2], hDD[2][2], aDD[0][0], aDD[0][1], aDD[0][2], aDD[1][1], aDD[1][2], aDD[2][2], trK, vetU[0], vetU[1], vetU[2], betU[0], betU[1], betU[2], alpha, cf], ["*hDD00", "*hDD01", "*hDD02", "*hDD11", "*hDD12", "*hDD22", "*aDD00", "*aDD01", "*aDD02", "*aDD11", "*aDD12", "*aDD22", "*trK", "*vetU0", "*vetU1", "*vetU2", "*betU0", "*betU1", "*betU2", "*alpha", "*cf"], "returnstring", params=outCparams), enableCparameters=enableCparameters) # Step 5.a: Output the driver function for the above # function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs() # Next write the driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(): desc = """Driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(), which writes BSSN variables in terms of ADM variables at a given point xx0,xx1,xx2""" name = "ID_BSSN__ALL_BUT_LAMBDAs" params = "const paramstruct *restrict params,REAL *restrict xx[3]," + ID_inputs_param + "REAL *in_gfs" enableCparameters = True funccallparams = "params, " idx3replace = "IDX3S" idx4ptreplace = "IDX4ptS" if "oldloops" in loopopts: params = "const int Nxx_plus_2NGHOSTS[3],REAL *xx[3]," + ID_inputs_param + "REAL *in_gfs" enableCparameters = False funccallparams = "" idx3replace = "IDX3" idx4ptreplace = "IDX4pt" outCfunction( outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params, body=""" const int idx = IDX3(i0,i1,i2); const int i0i1i2[3] = {i0,i1,i2}; const REAL xx0xx1xx2[3] = {xx0,xx1,xx2}; ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(""".replace("IDX3",idx3replace)+funccallparams+"""i0i1i2,xx0xx1xx2,other_inputs, &in_gfs[IDX4pt(HDD00GF,idx)],&in_gfs[IDX4pt(HDD01GF,idx)],&in_gfs[IDX4pt(HDD02GF,idx)], &in_gfs[IDX4pt(HDD11GF,idx)],&in_gfs[IDX4pt(HDD12GF,idx)],&in_gfs[IDX4pt(HDD22GF,idx)], &in_gfs[IDX4pt(ADD00GF,idx)],&in_gfs[IDX4pt(ADD01GF,idx)],&in_gfs[IDX4pt(ADD02GF,idx)], &in_gfs[IDX4pt(ADD11GF,idx)],&in_gfs[IDX4pt(ADD12GF,idx)],&in_gfs[IDX4pt(ADD22GF,idx)], &in_gfs[IDX4pt(TRKGF,idx)], &in_gfs[IDX4pt(VETU0GF,idx)],&in_gfs[IDX4pt(VETU1GF,idx)],&in_gfs[IDX4pt(VETU2GF,idx)], &in_gfs[IDX4pt(BETU0GF,idx)],&in_gfs[IDX4pt(BETU1GF,idx)],&in_gfs[IDX4pt(BETU2GF,idx)], &in_gfs[IDX4pt(ALPHAGF,idx)],&in_gfs[IDX4pt(CFGF,idx)]); """.replace("IDX4pt",idx4ptreplace), loopopts="AllPoints,Read_xxs"+loopopts, enableCparameters=enableCparameters)
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_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]
def Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear( CoordType_in, ADM_input_function_name, pointer_to_ID_inputs=False): # The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations. # To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in # the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3. # Step 0: Set spatial dimension (must be 3 for BSSN) DIM = 3 # Step 1: All ADM initial data quantities are now functions of xx0,xx1,xx2, but # they are still in the Spherical or Cartesian basis. We can now directly apply # Jacobian transformations to get them in the correct xx0,xx1,xx2 basis: # Step 1: All input quantities are in terms of r,th,ph or x,y,z. We want them in terms # of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace # r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2 # as defined for this particular reference metric in reference_metric.py's # xxSph[] or xxCart[], respectively: # Define the input variables: gammaSphorCartDD = ixp.declarerank2("gammaSphorCartDD", "sym01") KSphorCartDD = ixp.declarerank2("KSphorCartDD", "sym01") alphaSphorCart = sp.symbols("alphaSphorCart") betaSphorCartU = ixp.declarerank1("betaSphorCartU") BSphorCartU = ixp.declarerank1("BSphorCartU") # Make sure that rfm.reference_metric() has been called. # We'll need the variables it defines throughout this module. if rfm.have_already_called_reference_metric_function == False: print( "Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without" ) print( " first setting up reference metric, by calling rfm.reference_metric()." ) exit(1) r_th_ph_or_Cart_xyz_oID_xx = [] if CoordType_in == "Spherical": r_th_ph_or_Cart_xyz_oID_xx = rfm.xxSph elif CoordType_in == "Cartesian": r_th_ph_or_Cart_xyz_oID_xx = rfm.xxCart else: print( "Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords." ) exit(1) # Next apply Jacobian transformations to convert into the (xx0,xx1,xx2) basis # alpha is a scalar, so no Jacobian transformation is necessary. alpha = alphaSphorCart Jac_dUSphorCart_dDrfmUD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff( r_th_ph_or_Cart_xyz_oID_xx[i], rfm.xx[j]) Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3( Jac_dUSphorCart_dDrfmUD) betaU = ixp.zerorank1() BU = ixp.zerorank1() gammaDD = ixp.zerorank2() KDD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j] BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j] for k in range(DIM): for l in range(DIM): gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * \ gammaSphorCartDD[k][l] KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][ i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l] # Step 3: All ADM quantities were input into this function in the Spherical or Cartesian # basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above, # we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2. # Here we convert ADM quantities to their BSSN Curvilinear counterparts: # Step 3.1: Convert ADM $\gamma_{ij}$ to BSSN $\bar{\gamma}_{ij}$: # We have (Eqs. 2 and 3 of [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)): # \bar{\gamma}_{i j} = \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \gamma_{ij}. gammaUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaDD) gammabarDD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): gammabarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational( 1, 3)) * gammaDD[i][j] # Step 3.2: Convert the extrinsic curvature $K_{ij}$ to the trace-free extrinsic # curvature $\bar{A}_{ij}$, plus the trace of the extrinsic curvature $K$, # where (Eq. 3 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)): # K = \gamma^{ij} K_{ij}, and # \bar{A}_{ij} &= \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \left(K_{ij} - \frac{1}{3} \gamma_{ij} K \right) trK = sp.sympify(0) for i in range(DIM): for j in range(DIM): trK += gammaUU[i][j] * KDD[i][j] AbarDD = ixp.zerorank2() for i in range(DIM): for j in range(DIM): AbarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational( 1, 3)) * (KDD[i][j] - sp.Rational(1, 3) * gammaDD[i][j] * trK) # Step 3.3: Set the conformal factor variable $\texttt{cf}$, which is set # by the "BSSN_unrescaled_and_barred_vars::EvolvedConformalFactor_cf" parameter. For example if # "EvolvedConformalFactor_cf" is set to "phi", we can use Eq. 3 of # [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf), # which in arbitrary coordinates is written: # \phi = \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right). # Alternatively if "BSSN_unrescaled_and_barred_vars::EvolvedConformalFactor_cf" is set to "chi", then # \chi = e^{-4 \phi} = \exp\left(-4 \frac{1}{12} \left(\frac{\gamma}{\bar{\gamma}}\right)\right) # = \exp\left(-\frac{1}{3} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) = \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/3}. # # Finally if "BSSN_unrescaled_and_barred_vars::EvolvedConformalFactor_cf" is set to "W", then # W = e^{-2 \phi} = \exp\left(-2 \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) = # \exp\left(-\frac{1}{6} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) = # \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/6}. # First compute gammabarDET: gammabarUU, gammabarDET = ixp.symm_matrix_inverter3x3(gammabarDD) cf = sp.sympify(0) if par.parval_from_str("EvolvedConformalFactor_cf") == "phi": cf = sp.Rational(1, 12) * sp.log(gammaDET / gammabarDET) elif par.parval_from_str("EvolvedConformalFactor_cf") == "chi": cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 3)) elif par.parval_from_str("EvolvedConformalFactor_cf") == "W": cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 6)) else: print("Error EvolvedConformalFactor_cf type = \"" + par.parval_from_str("EvolvedConformalFactor_cf") + "\" unknown.") exit(1) # Step 4: Rescale tensorial quantities according to the prescription described in # the [BSSN in curvilinear coordinates tutorial module](Tutorial-BSSNCurvilinear.ipynb) # (also [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)): # # h_{ij} &= (\bar{\gamma}_{ij} - \hat{\gamma}_{ij})/\text{ReDD[i][j]}\\ # a_{ij} &= \bar{A}_{ij}/\text{ReDD[i][j]}\\ # \lambda^i &= \bar{\Lambda}^i/\text{ReU[i]}\\ # \mathcal{V}^i &= \beta^i/\text{ReU[i]}\\ # \mathcal{B}^i &= B^i/\text{ReU[i]}\\ hDD = ixp.zerorank2() aDD = ixp.zerorank2() 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] for j in range(DIM): hDD[i][j] = (gammabarDD[i][j] - rfm.ghatDD[i][j]) / rfm.ReDD[i][j] aDD[i][j] = AbarDD[i][j] / rfm.ReDD[i][j] # Step 5: Output all ADM-to-BSSN expressions to a C function. This function # must first call the ID_ADM_SphorCart() defined above. Using these # Spherical or Cartesian data, it sets up all quantities needed for # BSSNCurvilinear initial data, *except* $\lambda^i$, which must be # computed from numerical data using finite-difference derivatives. with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h", "w") as file: file.write( "void ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(const REAL xx0xx1xx2[3]," ) if pointer_to_ID_inputs == True: file.write("ID_inputs *other_inputs,") else: file.write("ID_inputs other_inputs,") file.write(""" REAL *hDD00,REAL *hDD01,REAL *hDD02,REAL *hDD11,REAL *hDD12,REAL *hDD22, REAL *aDD00,REAL *aDD01,REAL *aDD02,REAL *aDD11,REAL *aDD12,REAL *aDD22, REAL *trK, REAL *vetU0,REAL *vetU1,REAL *vetU2, REAL *betU0,REAL *betU1,REAL *betU2, REAL *alpha, REAL *cf) { REAL gammaSphorCartDD00,gammaSphorCartDD01,gammaSphorCartDD02, gammaSphorCartDD11,gammaSphorCartDD12,gammaSphorCartDD22; REAL KSphorCartDD00,KSphorCartDD01,KSphorCartDD02, KSphorCartDD11,KSphorCartDD12,KSphorCartDD22; REAL alphaSphorCart,betaSphorCartU0,betaSphorCartU1,betaSphorCartU2; REAL BSphorCartU0,BSphorCartU1,BSphorCartU2; const REAL xx0 = xx0xx1xx2[0]; const REAL xx1 = xx0xx1xx2[1]; const REAL xx2 = xx0xx1xx2[2]; REAL xyz_or_rthph[3];\n""") outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False" outputC(r_th_ph_or_Cart_xyz_oID_xx[0:3], ["xyz_or_rthph[0]", "xyz_or_rthph[1]", "xyz_or_rthph[2]"], "BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h", outCparams + ",CSE_enable=False") with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h", "a") as file: file.write(" " + ADM_input_function_name + """(xyz_or_rthph, other_inputs, &gammaSphorCartDD00,&gammaSphorCartDD01,&gammaSphorCartDD02, &gammaSphorCartDD11,&gammaSphorCartDD12,&gammaSphorCartDD22, &KSphorCartDD00,&KSphorCartDD01,&KSphorCartDD02, &KSphorCartDD11,&KSphorCartDD12,&KSphorCartDD22, &alphaSphorCart,&betaSphorCartU0,&betaSphorCartU1,&betaSphorCartU2, &BSphorCartU0,&BSphorCartU1,&BSphorCartU2); // Next compute all rescaled BSSN curvilinear quantities:\n""") outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False" outputC([ hDD[0][0], hDD[0][1], hDD[0][2], hDD[1][1], hDD[1][2], hDD[2][2], aDD[0][0], aDD[0][1], aDD[0][2], aDD[1][1], aDD[1][2], aDD[2][2], trK, vetU[0], vetU[1], vetU[2], betU[0], betU[1], betU[2], alpha, cf ], [ "*hDD00", "*hDD01", "*hDD02", "*hDD11", "*hDD12", "*hDD22", "*aDD00", "*aDD01", "*aDD02", "*aDD11", "*aDD12", "*aDD22", "*trK", "*vetU0", "*vetU1", "*vetU2", "*betU0", "*betU1", "*betU2", "*alpha", "*cf" ], "BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h", params=outCparams) with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h", "a") as file: file.write("}\n") # Step 5.A: Output the driver function for the above # function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs() # Next write the driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(): with open("BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h", "w") as file: file.write( "void ID_BSSN__ALL_BUT_LAMBDAs(const int Nxx_plus_2NGHOSTS[3],REAL *xx[3]," ) if pointer_to_ID_inputs == True: file.write("ID_inputs *other_inputs,") else: file.write("ID_inputs other_inputs,") file.write("REAL *in_gfs) {\n") file.write( lp.loop(["i2", "i1", "i0"], ["0", "0", "0"], [ "Nxx_plus_2NGHOSTS[2]", "Nxx_plus_2NGHOSTS[1]", "Nxx_plus_2NGHOSTS[0]" ], ["1", "1", "1"], [ "#pragma omp parallel for", " const REAL xx2 = xx[2][i2];", " const REAL xx1 = xx[1][i1];" ], "", """const REAL xx0 = xx[0][i0]; const int idx = IDX3(i0,i1,i2); const REAL xx0xx1xx2[3] = {xx0,xx1,xx2}; ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(xx0xx1xx2,other_inputs, &in_gfs[IDX4pt(HDD00GF,idx)],&in_gfs[IDX4pt(HDD01GF,idx)],&in_gfs[IDX4pt(HDD02GF,idx)], &in_gfs[IDX4pt(HDD11GF,idx)],&in_gfs[IDX4pt(HDD12GF,idx)],&in_gfs[IDX4pt(HDD22GF,idx)], &in_gfs[IDX4pt(ADD00GF,idx)],&in_gfs[IDX4pt(ADD01GF,idx)],&in_gfs[IDX4pt(ADD02GF,idx)], &in_gfs[IDX4pt(ADD11GF,idx)],&in_gfs[IDX4pt(ADD12GF,idx)],&in_gfs[IDX4pt(ADD22GF,idx)], &in_gfs[IDX4pt(TRKGF,idx)], &in_gfs[IDX4pt(VETU0GF,idx)],&in_gfs[IDX4pt(VETU1GF,idx)],&in_gfs[IDX4pt(VETU2GF,idx)], &in_gfs[IDX4pt(BETU0GF,idx)],&in_gfs[IDX4pt(BETU1GF,idx)],&in_gfs[IDX4pt(BETU2GF,idx)], &in_gfs[IDX4pt(ALPHAGF,idx)],&in_gfs[IDX4pt(CFGF,idx)]); """)) file.write("}\n") # Step 6: Compute $\bar{\Lambda}^i$ (Eqs. 4 and 5 of # [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)), # from finite-difference derivatives of rescaled metric # quantities $h_{ij}$: # \bar{\Lambda}^i = \bar{\gamma}^{jk}\left(\bar{\Gamma}^i_{jk} - \hat{\Gamma}^i_{jk}\right). # The reference_metric.py module provides us with analytic expressions for # $\hat{\Gamma}^i_{jk}$, so here we need only compute # finite-difference expressions for $\bar{\Gamma}^i_{jk}$, based on # the values for $h_{ij}$ provided in the initial data. Once # $\bar{\Lambda}^i$ has been computed, we apply the usual rescaling # procedure: # \lambda^i = \bar{\Lambda}^i/\text{ReU[i]}, # and then output the result to a C file using the NRPy+ # finite-difference C output routine. # We will need all BSSN gridfunctions to be defined, as well as # expressions for gammabarDD_dD in terms of exact derivatives of # the rescaling matrix and finite-difference derivatives of # hDD's. This functionality is provided by BSSN.BSSN_unrescaled_and_barred_vars, # which we call here to overwrite above definitions of gammabarDD,gammabarUU, etc. Bq.gammabar__inverse_and_derivs() # Provides gammabarUU and GammabarUDD gammabarUU = Bq.gammabarUU GammabarUDD = Bq.GammabarUDD # Next evaluate \bar{\Lambda}^i, based on GammabarUDD above and GammahatUDD # (from the reference metric): LambdabarU = ixp.zerorank1() for i in range(DIM): for j in range(DIM): for k in range(DIM): LambdabarU[i] += gammabarUU[j][k] * (GammabarUDD[i][j][k] - rfm.GammahatUDD[i][j][k]) # Finally apply rescaling: # lambda^i = Lambdabar^i/\text{ReU[i]} lambdaU = ixp.zerorank1() for i in range(DIM): lambdaU[i] = LambdabarU[i] / rfm.ReU[i] outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False" lambdaU_expressions = [ lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU0"), rhs=lambdaU[0]), lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU1"), rhs=lambdaU[1]), lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU2"), rhs=lambdaU[2]) ] lambdaU_expressions_FDout = fin.FD_outputC("returnstring", lambdaU_expressions, outCparams) with open("BSSN/ID_BSSN_lambdas.h", "w") as file: file.write(""" void ID_BSSN_lambdas(const int Nxx[3],const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],const REAL dxx[3],REAL *in_gfs) {\n""" ) file.write( lp.loop(["i2", "i1", "i0"], ["NGHOSTS", "NGHOSTS", "NGHOSTS"], ["NGHOSTS+Nxx[2]", "NGHOSTS+Nxx[1]", "NGHOSTS+Nxx[0]"], ["1", "1", "1"], [ "const REAL invdx0 = 1.0/dxx[0];\n" + "const REAL invdx1 = 1.0/dxx[1];\n" + "const REAL invdx2 = 1.0/dxx[2];\n" + "#pragma omp parallel for", " const REAL xx2 = xx[2][i2];", " const REAL xx1 = xx[1][i1];" ], "", "const REAL xx0 = xx[0][i0];\n" + lambdaU_expressions_FDout)) file.write("}\n")