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SpECTRE
v2024.04.12
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Ideal general relativistic magnetohydrodynamics (GRMHD) system with divergence cleaning coupled with electron fraction. More...
#include <System.hpp>
Public Types | |
| using | boundary_conditions_base = BoundaryConditions::BoundaryCondition |
| using | boundary_correction_base = BoundaryCorrections::BoundaryCorrection |
| using | variables_tag = ::Tags::Variables< tmpl::list< Tags::TildeD, Tags::TildeYe, Tags::TildeTau, Tags::TildeS<>, Tags::TildeB<>, Tags::TildePhi > > |
| using | flux_variables = tmpl::list< Tags::TildeD, Tags::TildeYe, Tags::TildeTau, Tags::TildeS<>, Tags::TildeB<>, Tags::TildePhi > |
| using | non_conservative_variables = tmpl::list<> |
| using | gradient_variables = tmpl::list<> |
| using | primitive_variables_tag = ::Tags::Variables< hydro::grmhd_tags< DataVector > > |
| using | spacetime_variables_tag = ::Tags::Variables< gr::tags_for_hydro< volume_dim, DataVector > > |
| using | flux_spacetime_variables_tag = ::Tags::Variables< tmpl::list< gr::Tags::Lapse< DataVector >, gr::Tags::Shift< DataVector, 3 >, gr::Tags::SpatialMetric< DataVector, 3 >, gr::Tags::SqrtDetSpatialMetric< DataVector >, gr::Tags::InverseSpatialMetric< DataVector, 3 > > > |
| using | compute_volume_time_derivative_terms = TimeDerivativeTerms |
| using | conservative_from_primitive = ConservativeFromPrimitive |
| template<typename OrderedListOfPrimitiveRecoverySchemes > | |
| using | primitive_from_conservative = PrimitiveFromConservative< OrderedListOfPrimitiveRecoverySchemes > |
| using | compute_largest_characteristic_speed = Tags::ComputeLargestCharacteristicSpeed |
| using | inverse_spatial_metric_tag = gr::Tags::InverseSpatialMetric< DataVector, volume_dim > |
Static Public Attributes | |
| static constexpr bool | is_in_flux_conservative_form = true |
| static constexpr bool | has_primitive_and_conservative_vars = true |
| static constexpr size_t | volume_dim = 3 |
Ideal general relativistic magnetohydrodynamics (GRMHD) system with divergence cleaning coupled with electron fraction.
We adopt the standard 3+1 form of metric
\begin{align*} ds^2 = -\alpha^2 dt^2 + \gamma_{ij}(dx^i + \beta^i dt)(dx^j + \beta^j dt) \end{align*}
where \(\alpha\) is the lapse, \(\beta^i\) is the shift vector, and \(\gamma_{ij}\) is the spatial metric.
Primitive variables of the system are
\begin{align*} v^i = \frac{1}{\alpha}\left(\frac{u^i}{u^0} + \beta^i\right) \end{align*}
measured by the Eulerian observerwith corresponding derived physical quantities which frequently appear in equations:
\begin{align*} W = -u^an_a = \alpha u^0 = \frac{1}{\sqrt{1-\gamma_{ij}v^iv^j}} = \sqrt{1+\gamma^{ij}u_iu_j} = \sqrt{1+\gamma^{ij}W^2v_iv_j} \end{align*}
\begin{align*} b^0 & = W B^i v_i / \alpha \\ b^i & = (B^i + \alpha b^0 u^i) / W \\ b^2 & = b^ab_a = B^2/W^2 + (B^iv_i)^2 \end{align*}
\begin{align*} \sqrt{4\pi}\frac{c^4}{G^{3/2}M_\odot} \approx 8.35 \times 10^{19} \,\text{Gauss} \end{align*}
For example, magnetic field \(10^{-5}\) with the code unit corresponds to the \(8.35\times 10^{14}\,\text{G}\) in the CGS Gaussian unit. See also documentation of hydro::units::cgs::gauss_unit for details.The GRMHD equations can be written in a flux-balanced form
\[ \partial_t U+ \partial_i F^i(U) = S(U). \]
Evolved (conserved) variables \(U\) are
\begin{align*} U = \sqrt{\gamma}\left[\,\begin{matrix} D \\ D Y_e \\ S_j \\ \tau \\ B^j \\ \Phi \\ \end{matrix}\,\right] \equiv \left[\,\,\begin{matrix} \tilde{D} \\ \tilde{Y_e} \\ \tilde{S}_j \\ \tilde{\tau} \\ \tilde{B}^j \\ \tilde{\Phi} \\ \end{matrix}\,\,\right] = \sqrt{\gamma} \left[\,\,\begin{matrix} \rho W \\ \rho W Y_e \\ (\rho h)^* W^2 v_j - \alpha b^0 b_j \\ (\rho h)^* W^2 - p^* - (\alpha b^0)^2 - \rho W \\ B^j \\ \Phi \\ \end{matrix}\,\,\right] \end{align*}
where \({\tilde D}\), \({\tilde Y}_e\), \({\tilde S}_i\), \({\tilde \tau}\), \({\tilde B}^i\), and \({\tilde \Phi}\) are a generalized mass-energy density, electron fraction, momentum density, specific internal energy density, magnetic field, and divergence cleaning field. Also, \(\gamma\) is the determinant of the spatial metric \(\gamma_{ij}\).
Corresponding fluxes \(F^i(U)\) are
\begin{align*} F^i({\tilde D}) &= {\tilde D} v^i_{tr} \\ F^i({\tilde Y}_e) &= {\tilde Y}_e v^i_{tr} \\ F^i({\tilde S}_j) &= {\tilde S}_j v^i_{tr} + \alpha \sqrt{\gamma} p^* \delta^i_j - \alpha b_j \tilde{B}^i / W \\ F^i({\tilde \tau}) &= {\tilde \tau} v^i_{tr} + \alpha \sqrt{\gamma} p^* v^i - \alpha^2 b^0 \tilde{B}^i / W \\ F^i({\tilde B}^j) &= {\tilde B}^j v^i_{tr} - \alpha v^j {\tilde B}^i + \alpha \gamma^{ij} {\tilde \Phi} \\ F^i({\tilde \Phi}) &= \alpha {\tilde B^i} - {\tilde \Phi} \beta^i \end{align*}
and source terms \(S(U)\) are
\begin{align*} S({\tilde D}) &= 0 \\ S({\tilde Y}_e) &= 0 \\ S({\tilde S}_j) &= \frac{\alpha}{2} {\tilde S}^{mn} \partial_j \gamma_{mn} + {\tilde S}_k \partial_j \beta^k - ({\tilde D} + {\tilde \tau}) \partial_j \alpha \\ S({\tilde \tau}) &= \alpha {\tilde S}^{mn} K_{mn} - {\tilde S}^m \partial_m \alpha \\ S({\tilde B}^j) &= -{\tilde B}^m \partial_m \beta^j + \Phi \partial_k (\alpha \sqrt{\gamma}\gamma^{jk}) \\ S({\tilde \Phi}) &= {\tilde B}^k \partial_k \alpha - \alpha K {\tilde \Phi} - \alpha \kappa {\tilde \Phi} \end{align*}
with
\begin{align*} {\tilde S}^{ij} = & \sqrt{\gamma} \left[ (\rho h)^* W^2 v^i v^j + p^* \gamma^{ij} - \gamma^{ik}\gamma^{jl}b_kb_l \right] \end{align*}
where \(K\) 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 \(\Phi = \partial_i {\tilde B}^i = 0\).