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SpECTRE
2021.08.02
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The Valencia formulation of ideal GRMHD with divergence cleaning. More...
Namespaces | |
| namespace | BoundaryConditions |
| Boundary conditions for the GRMHD Valencia Divergence Cleaning system. | |
| namespace | BoundaryCorrections |
| Boundary corrections/numerical fluxes. | |
| namespace | fd |
| Finite difference functionality for the ValenciaDivClean form of the GRMHD equations. | |
| namespace | PrimitiveRecoverySchemes |
| Schemes for recovering primitive variables from conservative variables. | |
| namespace | subcell |
| Code required by the DG-subcell/FD hybrid solver. | |
| namespace | Tags |
| Tags for the Valencia formulation of the ideal GRMHD equations with divergence cleaning. | |
Classes | |
| struct | ComputeFluxes |
| The fluxes of the conservative variables. More... | |
| struct | ComputeSources |
| Compute the source terms for the flux-conservative Valencia formulation of GRMHD with divergence cleaning. More... | |
| struct | ConservativeFromPrimitive |
| Compute the conservative variables from primitive variables. More... | |
| class | FixConservatives |
| Fix conservative variables using method developed by Foucart. More... | |
| struct | PrimitiveFromConservative |
| Compute the primitive variables from the conservative variables. More... | |
| struct | SetVariablesNeededFixingToFalse |
Mutator used with Initialization::Actions::AddSimpleTags to initialize the VariablesNeededFixing to false More... | |
| struct | System |
| struct | TimeDerivativeTerms |
| Compute the time derivative of the conserved variables for the Valencia formulation of the GRMHD equations with divergence cleaning. More... | |
Functions | |
| bool | operator!= (const FixConservatives &lhs, const FixConservatives &rhs) noexcept |
| template<size_t ThermodynamicDim> | |
| std::array< DataVector, 9 > | characteristic_speeds (const Scalar< DataVector > &rest_mass_density, const Scalar< DataVector > &specific_internal_energy, const Scalar< DataVector > &specific_enthalpy, const tnsr::I< DataVector, 3, Frame::Inertial > &spatial_velocity, const Scalar< DataVector > &lorentz_factor, const tnsr::I< DataVector, 3, Frame::Inertial > &magnetic_field, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, 3 > &shift, const tnsr::ii< DataVector, 3, Frame::Inertial > &spatial_metric, const tnsr::i< DataVector, 3 > &unit_normal, const EquationsOfState::EquationOfState< true, ThermodynamicDim > &equation_of_state) noexcept |
| Compute the characteristic speeds for the Valencia formulation of GRMHD with divergence cleaning. More... | |
| template<size_t ThermodynamicDim> | |
| void | characteristic_speeds (gsl::not_null< std::array< DataVector, 9 > * > char_speeds, const Scalar< DataVector > &rest_mass_density, const Scalar< DataVector > &specific_internal_energy, const Scalar< DataVector > &specific_enthalpy, const tnsr::I< DataVector, 3, Frame::Inertial > &spatial_velocity, const Scalar< DataVector > &lorentz_factor, const tnsr::I< DataVector, 3, Frame::Inertial > &magnetic_field, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, 3 > &shift, const tnsr::ii< DataVector, 3, Frame::Inertial > &spatial_metric, const tnsr::i< DataVector, 3 > &unit_normal, const EquationsOfState::EquationOfState< true, ThermodynamicDim > &equation_of_state) noexcept |
| Compute the characteristic speeds for the Valencia formulation of GRMHD with divergence cleaning. More... | |
The Valencia formulation of ideal GRMHD with divergence cleaning.
References:
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noexcept |
Compute the characteristic speeds for the Valencia formulation of GRMHD with divergence cleaning.
Obtaining the exact form of the characteristic speeds involves the solution of a nontrivial quartic equation for the fast and slow modes. Here we make use of a common approximation in the literature (e.g. [47]) where the resulting characteristic speeds are analogous to those of the Valencia formulation of the 3-D relativistic Euler system (see RelativisticEuler::Valencia::characteristic_speeds),
\begin{align*} \lambda_2 &= \alpha \Lambda^- - \beta_n,\\ \lambda_{3, 4, 5, 6, 7} &= \alpha v_n - \beta_n,\\ \lambda_{8} &= \alpha \Lambda^+ - \beta_n, \end{align*}
with the substitution
\begin{align*} c_s^2 \longrightarrow c_s^2 + v_A^2(1 - c_s^2) \end{align*}
in the definition of \(\Lambda^\pm\). Here \(v_A\) is the Alfvén speed. In addition, two more speeds corresponding to the divergence cleaning mode and the longitudinal magnetic field are added,
\begin{align*} \lambda_1 = -\alpha - \beta_n,\\ \lambda_9 = \alpha - \beta_n. \end{align*}
\begin{align*} v^2 &= \gamma_{mn} v^m v^n \\ c_s^2 &= \frac{1}{h} \left[ \left( \frac{\partial p}{\partial \rho} \right)_\epsilon + \frac{p}{\rho^2} \left(\frac{\partial p}{\partial \epsilon} \right)_\rho \right] \\ v_A^2 &= \frac{b^2}{b^2 + \rho h} \\ b^2 &= \frac{1}{W^2} \gamma_{mn} B^m B^n + \left( \gamma_{mn} B^m v^n \right)^2 \end{align*}
where \(\gamma_{mn}\) is the spatial metric, \(\rho\) is the rest mass density, \(W = 1/\sqrt{1-v_i v^i}\) is the Lorentz factor, \(h = 1 + \epsilon + \frac{p}{\rho}\) is the specific enthalpy, \(v^i\) is the spatial velocity, \(\epsilon\) is the specific internal energy, \(p\) is the pressure, and \(B^i\) is the spatial magnetic field measured by an Eulerian observer.
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noexcept |
Compute the characteristic speeds for the Valencia formulation of GRMHD with divergence cleaning.
Obtaining the exact form of the characteristic speeds involves the solution of a nontrivial quartic equation for the fast and slow modes. Here we make use of a common approximation in the literature (e.g. [47]) where the resulting characteristic speeds are analogous to those of the Valencia formulation of the 3-D relativistic Euler system (see RelativisticEuler::Valencia::characteristic_speeds),
\begin{align*} \lambda_2 &= \alpha \Lambda^- - \beta_n,\\ \lambda_{3, 4, 5, 6, 7} &= \alpha v_n - \beta_n,\\ \lambda_{8} &= \alpha \Lambda^+ - \beta_n, \end{align*}
with the substitution
\begin{align*} c_s^2 \longrightarrow c_s^2 + v_A^2(1 - c_s^2) \end{align*}
in the definition of \(\Lambda^\pm\). Here \(v_A\) is the Alfvén speed. In addition, two more speeds corresponding to the divergence cleaning mode and the longitudinal magnetic field are added,
\begin{align*} \lambda_1 = -\alpha - \beta_n,\\ \lambda_9 = \alpha - \beta_n. \end{align*}
\begin{align*} v^2 &= \gamma_{mn} v^m v^n \\ c_s^2 &= \frac{1}{h} \left[ \left( \frac{\partial p}{\partial \rho} \right)_\epsilon + \frac{p}{\rho^2} \left(\frac{\partial p}{\partial \epsilon} \right)_\rho \right] \\ v_A^2 &= \frac{b^2}{b^2 + \rho h} \\ b^2 &= \frac{1}{W^2} \gamma_{mn} B^m B^n + \left( \gamma_{mn} B^m v^n \right)^2 \end{align*}
where \(\gamma_{mn}\) is the spatial metric, \(\rho\) is the rest mass density, \(W = 1/\sqrt{1-v_i v^i}\) is the Lorentz factor, \(h = 1 + \epsilon + \frac{p}{\rho}\) is the specific enthalpy, \(v^i\) is the spatial velocity, \(\epsilon\) is the specific internal energy, \(p\) is the pressure, and \(B^i\) is the spatial magnetic field measured by an Eulerian observer.