ForceFree::Fluxes Struct Reference

Compute the fluxes of the GRFFE system with divergence cleaning. More...

#include <Fluxes.hpp>

Public Types
using	return_tags = implementation defined
using	argument_tags = implementation defined

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)

Detailed Description

Compute the fluxes of the GRFFE system with divergence cleaning.

$$\begin{array}{rl}{F}^j({\tilde{E}}^i)& =-{\beta }^j{\tilde{E}}^i+\alpha ({\gamma }^{ij}\tilde{\psi }-{ϵ}_{(3)}^{ijk}{\tilde{B}}_k)\\ F^j({\tilde{B}}^i)& =-{\beta }^j{\tilde{B}}^i+\alpha ({\gamma }^{ij}\tilde{\varphi }+{ϵ}_{(3)}^{ijk}{\tilde{E}}_k)\\ F^j(\tilde{\psi })& =-{\beta }^j\tilde{\psi }+\alpha {\tilde{E}}^j\\ F^j(\tilde{\varphi })& =-{\beta }^j\tilde{\varphi }+\alpha {\tilde{B}}^j\\ F^j(\tilde{q})& =\alpha \sqrt{\gamma }{J}^j-{\beta }^j\tilde{q}\end{array}$$

where the conserved variables ${\tilde{E}}^i,{\tilde{B}}^i,\tilde{\psi },\tilde{\varphi },\tilde{q}$ 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.

${ϵ}_{(3)}^{ijk}$ is the spatial Levi-Civita tensor defined as

$$\begin{array}{r}{ϵ}_{(3)}^{ijk}\equiv n_{\mu }{ϵ}^{\mu ijk}=-\frac{1}{\sqrt{-g}}{n}_{\mu }[\mu ijk]=\frac{1}{\sqrt{\gamma }}[ijk]\end{array}$$

where ${ϵ}^{\mu \nu \rho \sigma }$ 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

$$\begin{array}{r}{ϵ}_{\mu \nu \rho \sigma }=\sqrt{-g}[\mu \nu \rho \sigma ],\text{and}{ϵ}^{\mu \nu \rho \sigma }=-\frac{1}{\sqrt{-g}}[\mu \nu \rho \sigma ]\end{array}$$

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

• src/Evolution/Systems/ForceFree/Fluxes.hpp