SpECTRE  v2022.05.05
Ccz4::TimeDerivative< Dim > Struct Template Reference

Compute the RHS of the first order CCZ4 formulation of Einstein's equations [45]. More...

#include <TimeDerivative.hpp>

Static Public Member Functions

static void apply (const gsl::not_null< tnsr::ii< DataVector, Dim > * > dt_conformal_spatial_metric, const gsl::not_null< Scalar< DataVector > * > dt_ln_lapse, const gsl::not_null< tnsr::I< DataVector, Dim > * > dt_shift, const gsl::not_null< Scalar< DataVector > * > dt_ln_conformal_factor, const gsl::not_null< tnsr::ii< DataVector, Dim > * > dt_a_tilde, const gsl::not_null< Scalar< DataVector > * > dt_trace_extrinsic_curvature, const gsl::not_null< Scalar< DataVector > * > dt_theta, const gsl::not_null< tnsr::I< DataVector, Dim > * > dt_gamma_hat, const gsl::not_null< tnsr::I< DataVector, Dim > * > dt_b, const gsl::not_null< tnsr::i< DataVector, Dim > * > dt_field_a, const gsl::not_null< tnsr::iJ< DataVector, Dim > * > dt_field_b, const gsl::not_null< tnsr::ijj< DataVector, Dim > * > dt_field_d, const gsl::not_null< tnsr::i< DataVector, Dim > * > dt_field_p, const gsl::not_null< Scalar< DataVector > * > conformal_factor_squared, const gsl::not_null< Scalar< DataVector > * > det_conformal_spatial_metric, const gsl::not_null< tnsr::II< DataVector, Dim > * > inv_conformal_spatial_metric, const gsl::not_null< tnsr::II< DataVector, Dim > * > inv_spatial_metric, const gsl::not_null< Scalar< DataVector > * > lapse, const gsl::not_null< Scalar< DataVector > * > slicing_condition, const gsl::not_null< Scalar< DataVector > * > d_slicing_condition, const gsl::not_null< tnsr::II< DataVector, Dim > * > inv_a_tilde, const gsl::not_null< tnsr::ij< DataVector, Dim > * > a_tilde_times_field_b, const gsl::not_null< tnsr::ii< DataVector, Dim > * > a_tilde_minus_one_third_conformal_metric_times_trace_a_tilde, const gsl::not_null< Scalar< DataVector > * > contracted_field_b, const gsl::not_null< tnsr::ijK< DataVector, Dim > * > symmetrized_d_field_b, const gsl::not_null< tnsr::i< DataVector, Dim > * > contracted_symmetrized_d_field_b, const gsl::not_null< tnsr::ijk< DataVector, Dim > * > field_b_times_field_d, const gsl::not_null< tnsr::i< DataVector, Dim > * > field_d_up_times_a_tilde, const gsl::not_null< tnsr::I< DataVector, Dim > * > contracted_field_d_up, const gsl::not_null< tnsr::ij< DataVector, Dim > * > conformal_metric_times_field_b, const gsl::not_null< tnsr::ijk< DataVector, Dim > * > conformal_metric_times_symmetrized_d_field_b, const gsl::not_null< tnsr::ii< DataVector, Dim > * > conformal_metric_times_trace_a_tilde, const gsl::not_null< tnsr::i< DataVector, Dim > * > inv_conformal_metric_times_d_a_tilde, const gsl::not_null< tnsr::I< DataVector, Dim > * > gamma_hat_minus_contracted_conformal_christoffel, const gsl::not_null< tnsr::iJ< DataVector, Dim > * > d_gamma_hat_minus_contracted_conformal_christoffel, const gsl::not_null< tnsr::i< DataVector, Dim > * > contracted_christoffel_second_kind, const gsl::not_null< tnsr::ij< DataVector, Dim > * > contracted_d_conformal_christoffel_difference, const gsl::not_null< Scalar< DataVector > * > k_minus_2_theta_c, const gsl::not_null< Scalar< DataVector > * > k_minus_k0_minus_2_theta_c, const gsl::not_null< tnsr::ii< DataVector, Dim > * > lapse_times_a_tilde, const gsl::not_null< tnsr::ijj< DataVector, Dim > * > lapse_times_d_a_tilde, const gsl::not_null< tnsr::i< DataVector, Dim > * > lapse_times_field_a, const gsl::not_null< tnsr::ii< DataVector, Dim > * > lapse_times_conformal_spatial_metric, const gsl::not_null< Scalar< DataVector > * > lapse_times_slicing_condition, const gsl::not_null< Scalar< DataVector > * > lapse_times_ricci_scalar_plus_divergence_z4_constraint, const gsl::not_null< tnsr::I< DataVector, Dim > * > shift_times_deriv_gamma_hat, const gsl::not_null< tnsr::ii< DataVector, Dim > * > inv_tau_times_conformal_metric, const gsl::not_null< Scalar< DataVector > * > trace_a_tilde, const gsl::not_null< tnsr::iJJ< DataVector, Dim > * > field_d_up, const gsl::not_null< tnsr::Ijj< DataVector, Dim > * > conformal_christoffel_second_kind, const gsl::not_null< tnsr::iJkk< DataVector, Dim > * > d_conformal_christoffel_second_kind, const gsl::not_null< tnsr::Ijj< DataVector, Dim > * > christoffel_second_kind, const gsl::not_null< tnsr::ii< DataVector, Dim > * > spatial_ricci_tensor, const gsl::not_null< tnsr::ij< DataVector, Dim > * > grad_grad_lapse, const gsl::not_null< Scalar< DataVector > * > divergence_lapse, const gsl::not_null< tnsr::I< DataVector, Dim > * > contracted_conformal_christoffel_second_kind, const gsl::not_null< tnsr::iJ< DataVector, Dim > * > d_contracted_conformal_christoffel_second_kind, const gsl::not_null< tnsr::i< DataVector, Dim > * > spatial_z4_constraint, const gsl::not_null< Scalar< DataVector > * > upper_spatial_z4_constraint_buffer, const gsl::not_null< tnsr::I< DataVector, Dim > * > upper_spatial_z4_constraint, const gsl::not_null< tnsr::ij< DataVector, Dim > * > grad_spatial_z4_constraint, const gsl::not_null< Scalar< DataVector > * > ricci_scalar_plus_divergence_z4_constraint, const double c, const double cleaning_speed, const Scalar< DataVector > &eta, const double f, const Scalar< DataVector > &k_0, const tnsr::i< DataVector, Dim > &d_k_0, const double kappa_1, const double kappa_2, const double kappa_3, const double mu, const double one_over_relaxation_time, const EvolveShift evolve_shift, const SlicingConditionType slicing_condition_type, const tnsr::ii< DataVector, Dim > &conformal_spatial_metric, const Scalar< DataVector > &ln_lapse, const tnsr::I< DataVector, Dim > &shift, const Scalar< DataVector > &ln_conformal_factor, const tnsr::ii< DataVector, Dim > &a_tilde, const Scalar< DataVector > &trace_extrinsic_curvature, const Scalar< DataVector > &theta, const tnsr::I< DataVector, Dim > &gamma_hat, const tnsr::I< DataVector, Dim > &b, const tnsr::i< DataVector, Dim > &field_a, const tnsr::iJ< DataVector, Dim > &field_b, const tnsr::ijj< DataVector, Dim > &field_d, const tnsr::i< DataVector, Dim > &field_p, const tnsr::ijj< DataVector, Dim > &d_a_tilde, const tnsr::i< DataVector, Dim > &d_trace_extrinsic_curvature, const tnsr::i< DataVector, Dim > &d_theta, const tnsr::iJ< DataVector, Dim > &d_gamma_hat, const tnsr::iJ< DataVector, Dim > &d_b, const tnsr::ij< DataVector, Dim > &d_field_a, const tnsr::ijK< DataVector, Dim > &d_field_b, const tnsr::ijkk< DataVector, Dim > &d_field_d, const tnsr::ij< DataVector, Dim > &d_field_p)
 

Detailed Description

template<size_t Dim>
struct Ccz4::TimeDerivative< Dim >

Compute the RHS of the first order CCZ4 formulation of Einstein's equations [45].

Details

We define \(\phi = (\det(\gamma_{ij}))^{-1/6}\) as the conformal factor, \(\alpha\) as the lapse, \(\beta^i\) as the shift, \(K_{ij}\) as the extrinsic curvature, and \(Z_{a}\) as the Z4 constraint.

The evolved variables are the conformal spatial metric \(\tilde{\gamma}_{ij} = \phi^2 \gamma_{ij}\), the natural log of the lapse \(\ln \alpha\), the shift \(\beta^i\), the natural log of the conformal factor \(\ln \phi\), the trace-free part of the extrinsic curvature \(\tilde A_{ij} = \phi^2 \left(K_{ij} - \frac{1}{3} K \gamma_{ij}\right)\), the trace of the extrinsic curvature \(K = K_{ij} \gamma^{ij}\), the projection of the Z4 four-vector along the normal direction \(\Theta = Z^0 \alpha\), \(\hat{\Gamma}^{i}\) defined by Ccz4::Tags::GammaHat, the free variable \(b^i\) that controls the evolution of the shift and its time derivative, the auxiliary variable \(A_i = \partial_i \ln(\alpha) = \frac{\partial_i \alpha}{\alpha}\), the auxiliary variable \(B_k{}^{i} = \partial_k \beta^i\), the auxiliary variable \(D_{kij} = \frac{1}{2} \partial_k \tilde{\gamma}_{ij}\), and the auxiliary variable \(P_i = \partial_i \ln(\phi) = \frac{\partial_i \phi}{\phi}\).

The evolution equations are equations 12a - 12m of [45] . Equations 13 - 27 define identities used in the evolution equations.

This evolution uses two settings that can be toggled:

  • evolve_shift governs whether or not the shift is evolved by setting \(s = 1\) or \(s = 0\)
  • slicing_condition_type governs which slicing condition to use by setting the value of \(g(\alpha)\))

The documentation for this struct was generated from the following file: