grmhd::ValenciaDivClean::System Struct Reference

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 = implementation defined
using	non_conservative_variables = implementation defined
using	gradient_variables = implementation defined
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

Detailed Description

Ideal general relativistic magnetohydrodynamics (GRMHD) system with divergence cleaning coupled with electron fraction.

We adopt the standard 3+1 form of metric

$$\begin{array}{r}d{s}^2=-{\alpha }^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+{\beta }^{i}dt)(d{x}^j+{\beta }^{j}dt)\end{array}$$

where $\alpha$ is the lapse, ${\beta }^i$ is the shift vector, and ${\gamma }_{ij}$ is the spatial metric.

Primitive variables of the system are

• rest mass density $\rho$
• electron fraction $Y_e$
• the spatial velocity

  $$\begin{array}{r}{v}^i=\frac{1}{\alpha }(\frac{u^i}{u^0}+{\beta }^i)\end{array}$$

  measured by the Eulerian observer
• specific internal energy density $ϵ$
• magnetic field $B^i={-}^{\ast }{F}^{ia}{n}_a=-\alpha {(}^{\ast }{F}^{0i})$
• the divergence cleaning field $\mathrm{\Phi }$

with corresponding derived physical quantities which frequently appear in equations:

• The transport velocity $v_{tr}^i=\alpha v^i-{\beta }^i$
• The Lorentz factor

  $$\begin{array}{r}W=-u^a{n}_a=\alpha u^0=\frac{1}{\sqrt{1-{\gamma }_{ij}{v}^i{v}^j}}=\sqrt{1+{\gamma }^{ij}{u}_i{u}_j}=\sqrt{1+{\gamma }^{ij}{W}^2{v}_i{v}_j}\end{array}$$

• The specific enthalpy $h=1+ϵ+p/\rho$ where $p$ is the pressure specified by a particular equation of state (EoS)
• The comoving magnetic field $b^b={-}^{\ast }{F}^{ba}{u}_a$ in the ideal MHD limit

  $$\begin{array}{rl}{b}^0& =W{B}^i{v}_i/\alpha \\ b^i& =(B^i+\alpha b^0{u}^i)/W\\ b^2& =b^a{b}_a=B^2/W^2+(B^i{v}_i{)}^2\end{array}$$

• Augmented enthalpy density $(\rho h{)}^{\ast }=\rho h+b^2$ and augmented pressure $p^{\ast }=p+b^2/2$ which include contributions from magnetic field

Note

We are using the electromagnetic variables with the scaling convention that the factor $4\pi$ does not appear in Maxwell equations and the stress-energy tensor of the EM fields (geometrized Heaviside-Lorentz units). To recover the physical value of magnetic field in the usual CGS Gaussian unit, the conversion factor is

$$\begin{array}{r}\sqrt{4\pi }\frac{c^4}{G^{3/2}{M}_{\odot }}\approx 8.35\times {10}^{19}\text{Gauss}\end{array}$$

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

$$\begin{array}{r}U=\sqrt{\gamma }\left[\begin{array}{c}D\\ D{Y}_e\\ S_j\\ \tau \\ B^j\\ \mathrm{\Phi }\end{array}\right]\equiv \left[\begin{array}{c}\tilde{D}\\ \widetilde{Y_e}\\ {\tilde{S}}_j\\ \tilde{\tau }\\ {\tilde{B}}^j\\ \tilde{\mathrm{\Phi }}\end{array}\right]=\sqrt{\gamma }\left[\begin{array}{c}\rho W\\ \rho W{Y}_e\\ (\rho h{)}^{\ast }{W}^2{v}_j-\alpha b^0{b}_j\\ (\rho h{)}^{\ast }{W}^2-p^{\ast }-(\alpha b^0{)}^2-\rho W\\ B^j\\ \mathrm{\Phi }\end{array}\right]\end{array}$$

where $\tilde{D}$, ${\tilde{Y}}_e$, ${\tilde{S}}_i$, $\tilde{\tau }$, ${\tilde{B}}^i$, and $\tilde{\mathrm{\Phi }}$ 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

$$\begin{array}{rl}{F}^i(\tilde{D})& =\tilde{D}{v}_{tr}^i\\ F^i({\tilde{Y}}_e)& ={\tilde{Y}}_e{v}_{tr}^i\\ F^i({\tilde{S}}_j)& ={\tilde{S}}_j{v}_{tr}^i+\alpha \sqrt{\gamma }{p}^{\ast }{\delta }_j^i-\alpha b_j{\tilde{B}}^i/W\\ F^i(\tilde{\tau })& =\tilde{\tau }{v}_{tr}^i+\alpha \sqrt{\gamma }{p}^{\ast }{v}^i-{\alpha }^2{b}^0{\tilde{B}}^i/W\\ F^i({\tilde{B}}^j)& ={\tilde{B}}^j{v}_{tr}^i-v_{tr}^j{\tilde{B}}^i+\alpha {\gamma }^{ij}\tilde{\mathrm{\Phi }}\\ F^i(\tilde{\mathrm{\Phi }})& =\alpha {\tilde{B}}^i-\tilde{\mathrm{\Phi }}{\beta }^i\end{array}$$

and source terms $S(U)$ are

$$\begin{array}{rl}S(\tilde{D})& =0\\ S({\tilde{Y}}_e)& =0\\ S({\tilde{S}}_j)& =\frac{\alpha }{2}{\tilde{S}}^{mn}{\partial }_j{\gamma }_{mn}+{\tilde{S}}_k{\partial }_j{\beta }^k-(\tilde{D}+\tilde{\tau }){\partial }_j\alpha \\ S(\tilde{\tau })& =\alpha {\tilde{S}}^{mn}{K}_{mn}-{\tilde{S}}^m{\partial }_m\alpha \\ S({\tilde{B}}^j)& =\mathrm{\Phi }{\partial }_k(\alpha \sqrt{\gamma }{\gamma }^{jk})\\ S(\tilde{\mathrm{\Phi }})& ={\tilde{B}}^k{\partial }_k\alpha -\alpha K\tilde{\mathrm{\Phi }}-\alpha \kappa \tilde{\mathrm{\Phi }}\end{array}$$

with

$$\begin{array}{rl}{\tilde{S}}^{ij}=& \sqrt{\gamma }[(\rho h{)}^{\ast }{W}^2{v}^i{v}^j+p^{\ast }{\gamma }^{ij}-{\gamma }^{ik}{\gamma }^{jl}{b}_k{b}_l]\end{array}$$

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 $\mathrm{\Phi }={\partial }_i{\tilde{B}}^i=0$.

Note

The fluxes and sources of $\tilde{B}$ follow eq. 32 of [159] rather than eq. 87 of [147]

On the electron fraction side, the source term is currently set to $S({\tilde{Y}}_e)=0$. Implementing the source term using neutrino scheme is in progress (Last update : Oct 2022).

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

• src/Evolution/Systems/GrMhd/ValenciaDivClean/System.hpp