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SpECTRE
v2024.04.12
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Compute the fluxes of the GRFFE system with divergence cleaning. More...
#include <Fluxes.hpp>
Public Types | |
| using | return_tags = tmpl::list<::Tags::Flux< Tags::TildeE, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::Flux< Tags::TildeB, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::Flux< Tags::TildePsi, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::Flux< Tags::TildePhi, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::Flux< Tags::TildeQ, tmpl::size_t< 3 >, Frame::Inertial > > |
| using | argument_tags = tmpl::list< Tags::TildeE, Tags::TildeB, Tags::TildePsi, Tags::TildePhi, Tags::TildeQ, Tags::TildeJ, gr::Tags::Lapse< DataVector >, gr::Tags::Shift< DataVector, 3 >, gr::Tags::SqrtDetSpatialMetric< DataVector >, gr::Tags::SpatialMetric< DataVector, 3 >, gr::Tags::InverseSpatialMetric< DataVector, 3 > > |
Static Public Member Functions | |
| static void | apply (gsl::not_null< tnsr::IJ< DataVector, 3, Frame::Inertial > * > tilde_e_flux, gsl::not_null< tnsr::IJ< DataVector, 3, Frame::Inertial > * > tilde_b_flux, gsl::not_null< tnsr::I< DataVector, 3, Frame::Inertial > * > tilde_psi_flux, gsl::not_null< tnsr::I< DataVector, 3, Frame::Inertial > * > tilde_phi_flux, gsl::not_null< tnsr::I< DataVector, 3, Frame::Inertial > * > tilde_q_flux, const tnsr::I< DataVector, 3, Frame::Inertial > &tilde_e, const tnsr::I< DataVector, 3, Frame::Inertial > &tilde_b, const Scalar< DataVector > &tilde_psi, const Scalar< DataVector > &tilde_phi, const Scalar< DataVector > &tilde_q, const tnsr::I< DataVector, 3, Frame::Inertial > &tilde_j, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, 3, Frame::Inertial > &shift, const Scalar< DataVector > &sqrt_det_spatial_metric, const tnsr::ii< DataVector, 3, Frame::Inertial > &spatial_metric, const tnsr::II< DataVector, 3, Frame::Inertial > &inv_spatial_metric) |
Compute the fluxes of the GRFFE system with divergence cleaning.
\begin{align*} F^j(\tilde{E}^i) & = -\beta^j\tilde{E}^i + \alpha (\gamma^{ij}\tilde{\psi} - \epsilon^{ijk}_{(3)}\tilde{B}_k) \\ F^j(\tilde{B}^i) & = -\beta^j\tilde{B}^i + \alpha (\gamma^{ij}\tilde{\phi} + \epsilon^{ijk}_{(3)}\tilde{E}_k) \\ F^j(\tilde{\psi}) & = -\beta^j \tilde{\psi} + \alpha \tilde{E}^j \\ F^j(\tilde{\phi}) & = -\beta^j \tilde{\phi} + \alpha \tilde{B}^j \\ F^j(\tilde{q}) & = \alpha \sqrt{\gamma}J^j - \beta^j \tilde{q} \end{align*}
where the conserved variables \(\tilde{E}^i, \tilde{B}^i, \tilde{\psi}, \tilde{\phi}, \tilde{q}\) are densitized electric field, magnetic field, electric divergence cleaning field, magnetic divergence cleaning field, and electric charge density. \(J^i\) is the spatial electric current density.
\(\epsilon_{(3)}^{ijk}\) is the spatial Levi-Civita tensor defined as
\begin{align*} \epsilon_{(3)}^{ijk} \equiv n_\mu \epsilon^{\mu ijk} = -\frac{1}{\sqrt{-g}} n_\mu [\mu ijk] = \frac{1}{\sqrt{\gamma}} [ijk] \end{align*}
where \(\epsilon^{\mu\nu\rho\sigma}\) is the Levi-Civita tensor, \(g\) is the determinant of spacetime metric, \(\gamma\) is the determinant of spatial metric, \(n^\mu\) is the normal to spatial hypersurface. Also, \([abcd]\) and \([ijk]\) are the usual antisymmetric symbols (which only have the value \(\pm 1\)) with 4 and 3 indices, respectively, with the sign \([0123] = [123] = +1\). Note that
\begin{align*} \epsilon_{\mu\nu\rho\sigma} = \sqrt{-g} \, [\mu\nu\rho\sigma] , \quad \text{and} \quad \epsilon^{\mu\nu\rho\sigma} = -\frac{1}{\sqrt{-g}} [\mu\nu\rho\sigma] \end{align*}