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SpECTRE
v2024.04.12
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Compute the source terms for the flux-conservative Valencia formulation of GRMHD with divergence cleaning, coupled with electron fraction. More...
#include <Sources.hpp>
Static Public Member Functions | |
| static void | apply (gsl::not_null< Scalar< DataVector > * > source_tilde_tau, gsl::not_null< tnsr::i< DataVector, 3, Frame::Inertial > * > source_tilde_s, gsl::not_null< tnsr::I< DataVector, 3, Frame::Inertial > * > source_tilde_b, gsl::not_null< Scalar< DataVector > * > source_tilde_phi, const Scalar< DataVector > &tilde_d, const Scalar< DataVector > &tilde_ye, const Scalar< DataVector > &tilde_tau, const tnsr::i< DataVector, 3, Frame::Inertial > &tilde_s, const tnsr::I< DataVector, 3, Frame::Inertial > &tilde_b, const Scalar< DataVector > &tilde_phi, const tnsr::I< DataVector, 3, Frame::Inertial > &spatial_velocity, const tnsr::I< DataVector, 3, Frame::Inertial > &magnetic_field, const Scalar< DataVector > &rest_mass_density, const Scalar< DataVector > &electron_fraction, const Scalar< DataVector > &specific_internal_energy, const Scalar< DataVector > &lorentz_factor, const Scalar< DataVector > &pressure, const Scalar< DataVector > &lapse, const tnsr::i< DataVector, 3, Frame::Inertial > &d_lapse, const tnsr::iJ< DataVector, 3, Frame::Inertial > &d_shift, const tnsr::ii< DataVector, 3, Frame::Inertial > &spatial_metric, const tnsr::ijj< DataVector, 3, Frame::Inertial > &d_spatial_metric, const tnsr::II< DataVector, 3, Frame::Inertial > &inv_spatial_metric, const Scalar< DataVector > &sqrt_det_spatial_metric, const tnsr::ii< DataVector, 3, Frame::Inertial > &extrinsic_curvature, double constraint_damping_parameter) |
Compute the source terms for the flux-conservative Valencia formulation of GRMHD with divergence cleaning, coupled with electron fraction.
A flux-conservative system has the generic form:
\[ \partial_t U_i + \partial_m F^m(U_i) = S(U_i) \]
where \(F^a()\) denotes the flux of a conserved variable \(U_i\) and \(S()\) denotes the source term for the conserved variable.
For the Valencia formulation:
\begin{align*} S({\tilde D}) = & 0\\ S({\tilde S}_i) = & \frac{1}{2} \alpha {\tilde S}^{mn} \partial_i \gamma_{mn} + {\tilde S}_m \partial_i \beta^m - ({\tilde D} + {\tilde \tau}) \partial_i \alpha \\ S({\tilde \tau}) = & \alpha {\tilde S}^{mn} K_{mn} - {\tilde S}^m \partial_m \alpha \\ S({\tilde B}^i) = & -{\tilde B}^m \partial_m \beta^i + {\tilde \Phi} \gamma^{im} \partial_m \alpha + \alpha {\tilde \Phi} \left( \frac{1}{2} \gamma^{il} \gamma^{jk} - \gamma^{ij} \gamma^{lk} \right) \partial_l \gamma_{jk} \\ S({\tilde \Phi}) = & {\tilde B}^k \partial_k \alpha - \alpha K {\tilde \Phi} - \alpha \kappa {\tilde \Phi} \end{align*}
where
\begin{align*} {\tilde S}^i = & {\tilde S}_m \gamma^{im} \\ {\tilde S}^{ij} = & \sqrt{\gamma} \left[ \left(h \rho W^2 + B^n B_n \right) v^i v^j + \left(p + p_m \right) \gamma^{ij} - B^n v_n \left( B^i v^j + B^j v^i \right) - \frac{B^i B^j}{W^2} \right] \end{align*}
where \({\tilde D}\), \({\tilde S}_i\), \({\tilde \tau}\), \({\tilde B}^i\), and \({\tilde \Phi}\) are a generalized mass-energy density, momentum density, specific internal energy density, magnetic field, and divergence cleaning field. Furthermore, \(\gamma\) is the determinant of the spatial metric \(\gamma_{ij}\), \(\rho\) is the rest mass density, \(W\) is the Lorentz factor, \(h\) is the specific enthalpy, \(v^i\) is the spatial velocity, \(B^k\) is the magnetic field, \(p\) is the pressure, \(p_m = \frac{1}{2} \left[ \left( B^n v_n \right)^2 + B^n B_n / W^2 \right]\) is the magnetic pressure, \(\alpha\) is the lapse, \(\beta^i\) is the shift, \(K\) is the trace of the extrinsic curvature \(K_{ij}\), and \(\kappa\) is a damping parameter that damps violations of the divergence-free (no-monopole) condition \(\Phi = \partial_i {\tilde B}^i = 0\) .