Items related to the Valencia formulation of ideal GRMHD with divergence cleaning coupled with electron fraction. 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, coupled with electron fraction. More...

struct	ConservativeFromPrimitive
	Compute the conservative variables from primitive variables. More...

class	FixConservatives
	Fix conservative variables using method developed by Foucart. More...

class	Flattener
	Reduces oscillations inside an element in an attempt to guarantee a physical solution of the conserved variables for which the primitive variables can be recovered. More...

class	NumericInitialData
	Numeric initial data loaded from volume data files. More...

struct	PrimitiveFromConservative
	Compute the primitive variables from the conservative variables. More...

class	PrimitiveFromConservativeOptions
	Options to be passed to the Con2Prim algorithm. Currently, we simply set a threshold for tildeD below which the inversion is not performed and the density is set to atmosphere values. More...

struct	SetVariablesNeededFixingToFalse
	Mutator used with `Initialization::Actions::AddSimpleTags` to initialize the `VariablesNeededFixing` to `false` More...

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

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)

template<typename RecoverySchemesList >
bool	operator!= (const Flattener< RecoverySchemesList > &lhs, const Flattener< RecoverySchemesList > &rhs)

bool	operator!= (const PrimitiveFromConservativeOptions &lhs, const PrimitiveFromConservativeOptions &rhs)


template<size_t ThermodynamicDim>
std::array< DataVector, 9 >	characteristic_speeds (const Scalar< DataVector > &rest_mass_density, const Scalar< DataVector > &electron_fraction, 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)
	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 > &electron_fraction, 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)
	Compute the characteristic speeds for the Valencia formulation of GRMHD with divergence cleaning. More...

Detailed Description

Items related to the Valencia formulation of ideal GRMHD with divergence cleaning coupled with electron fraction.

References:

Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Approach [4]
GRHydro: a new open-source general-relativistic magnetohydrodynamics code for the Einstein toolkit [136]
Black hole-neutron star mergers with a hot nuclear equation of state: outflow and neutrino-cooled disk for a low-mass, high-spin case [45]

Function Documentation

◆ characteristic_speeds() [1/2]

template<size_t ThermodynamicDim>

std::array< DataVector, 9 > grmhd::ValenciaDivClean::characteristic_speeds	(	const Scalar< DataVector > &	rest_mass_density,
		const Scalar< DataVector > &	electron_fraction,
		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
	)

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. [73]) 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*}

Note: The ordering assumed here is such that, in the Newtonian limit, the exact expressions for \(\lambda_{2, 8}\), \(\lambda_{3, 7}\), and \(\lambda_{4, 6}\) should reduce to the corresponding fast modes, Alfvén modes, and slow modes, respectively. See [46] for a detailed description of the hyperbolic characterization of Newtonian MHD. In terms of the primitive variables:

\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.

◆ characteristic_speeds() [2/2]