grmhd::GhValenciaDivClean Namespace Reference

Namespace associated with utilities for the combined Generalized Harmonic and Valencia formulation of ideal GRMHD with divergence cleaning systems. More...

Namespaces
namespace	BoundaryConditions
	Boundary conditions for the combined Generalized Harmonic and Valencia GRMHD systems.
namespace	BoundaryCorrections
	Boundary corrections/numerical fluxes.
namespace	fd
	Finite difference functionality for the coupled Generalized Harmonic and ValenciaDivClean equations.
namespace	subcell
	Code required by the DG-subcell/FD hybrid solver.
namespace	Tags
	Tags for the combined system of the Generalized Harmonic formulation for the Einstein field equations and the Valencia GRMHD formulation.

Classes
class	NumericInitialData
	Numeric initial data loaded from volume data files. More...
struct	SetPiAndPhiFromConstraints
	Set \(\Pi_{ab}\) from the gauge source function. More...
struct	System
struct	TimeDerivativeTerms
	Compute the RHS terms and flux values for both the Generalized Harmonic formulation of Einstein's equations and the Valencia formulation of the GRMHD equations with divergence cleaning. More...

Functions
void	add_stress_energy_term_to_dt_pi (gsl::not_null< tnsr::aa< DataVector, 3 > * > dt_pi, const tnsr::aa< DataVector, 3 > &trace_reversed_stress_energy, const Scalar< DataVector > &lapse)
	Add in the trace-reversed stress-energy source term to the \(\Pi\) evolved variable of the Generalized Harmonic system. More...
void	trace_reversed_stress_energy (gsl::not_null< tnsr::aa< DataVector, 3 > * > stress_energy, gsl::not_null< tnsr::a< DataVector, 3 > * > four_velocity_one_form_buffer, gsl::not_null< tnsr::a< DataVector, 3 > * > comoving_magnetic_field_one_form_buffer, const Scalar< DataVector > &rest_mass_density, const tnsr::i< DataVector, 3, Frame::Inertial > &spatial_velocity_one_form, const tnsr::i< DataVector, 3, Frame::Inertial > &magnetic_field_one_form, const Scalar< DataVector > &magnetic_field_squared, const Scalar< DataVector > &magnetic_field_dot_spatial_velocity, const Scalar< DataVector > &lorentz_factor, const Scalar< DataVector > &one_over_w_squared, const Scalar< DataVector > &pressure, const Scalar< DataVector > &specific_internal_energy, const tnsr::aa< DataVector, 3, Frame::Inertial > &spacetime_metric, const tnsr::I< DataVector, 3, Frame::Inertial > &shift, const Scalar< DataVector > &lapse)
	Calculate the trace-reversed stress-energy tensor \((T_{\mu \nu} - 1/2 g_{\mu \nu} g^{\lambda \sigma} T_{\lambda \sigma}) \) associated with the matter part of the GRMHD system. More...

Detailed Description

Namespace associated with utilities for the combined Generalized Harmonic and Valencia formulation of ideal GRMHD with divergence cleaning systems.

Function Documentation

add_stress_energy_term_to_dt_pi()

void grmhd::GhValenciaDivClean::add_stress_energy_term_to_dt_pi	(	gsl::not_null< tnsr::aa< DataVector, 3 > * >	dt_pi,
		const tnsr::aa< DataVector, 3 > &	trace_reversed_stress_energy,
		const Scalar< DataVector > &	lapse
	)

Add in the trace-reversed stress-energy source term to the \(\Pi\) evolved variable of the Generalized Harmonic system.

Details

The only stress energy source term in the Generalized Harmonic evolution equations is in the equation for \(\Pi_{a b}\):

\[ \partial_t \Pi_{ab} + \text{(spatial derivative terms)} = \text{(GH source terms)} - 16 \pi \alpha (T_{ab} - \frac{1}{2} g_{a b} T^c{}_c) \]

(note that this function takes as argument the trace-reversed stress energy tensor)

This function adds that contribution to the existing value of dt_pi. The spacetime terms in the GH equation should be computed before passing the dt_pi to this function for updating.

See also

gh::TimeDerivative for details about the spacetime part of the time derivative calculation.

trace_reversed_stress_energy()

void grmhd::GhValenciaDivClean::trace_reversed_stress_energy	(	gsl::not_null< tnsr::aa< DataVector, 3 > * >	stress_energy,
		gsl::not_null< tnsr::a< DataVector, 3 > * >	four_velocity_one_form_buffer,
		gsl::not_null< tnsr::a< DataVector, 3 > * >	comoving_magnetic_field_one_form_buffer,
		const Scalar< DataVector > &	rest_mass_density,
		const tnsr::i< DataVector, 3, Frame::Inertial > &	spatial_velocity_one_form,
		const tnsr::i< DataVector, 3, Frame::Inertial > &	magnetic_field_one_form,
		const Scalar< DataVector > &	magnetic_field_squared,
		const Scalar< DataVector > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataVector > &	lorentz_factor,
		const Scalar< DataVector > &	one_over_w_squared,
		const Scalar< DataVector > &	pressure,
		const Scalar< DataVector > &	specific_internal_energy,
		const tnsr::aa< DataVector, 3, Frame::Inertial > &	spacetime_metric,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	shift,
		const Scalar< DataVector > &	lapse
	)

Calculate the trace-reversed stress-energy tensor \((T_{\mu \nu} - 1/2 g_{\mu \nu} g^{\lambda \sigma} T_{\lambda \sigma}) \) associated with the matter part of the GRMHD system.

Details

The stress energy tensor is needed to compute the backreaction of the matter to the spacetime degrees of freedom. The stress energy is calculated as described in [133] :

\[ T_{\mu \nu} = \rho h^* u_\mu u_\nu + p^* g_{\mu \nu} - b_\mu b_\nu, \]

where \(u_\mu\) is the four-velocity, \(\rho\) is the rest mass density, \(h^*\) is the magnetically modified enthalpy, \(p^*\) is the magnetically modified pressure, and \(b_{\mu}\) is the comoving magnetic field (note that we deviate from the notation of [133] by denoting the pressure with a lower-case \(p\) instead of an upper-case \(P\)).

The spatial components of the four velocity \(u_\mu\) are

\[ u_i = W v_i, \]

and the time component is

\[ u_0 = - \alpha W + \beta^i u_i. \]

.

The magnetically modified enthalpy is

\[ h^* = 1 + \epsilon + (p + b^2) / \rho. \]

The magnetically modified pressure is

\[ p^* = p + b^2 / 2. \]

The comoving magnetic field is computed via

\begin{align} b_i &= B_i / W + v_i W v^k B_k\\ b_0 &= - \alpha W v^i B_i + \beta^i b_i \end{align}

Therefore, the trace-reversed stress energy simplifies to

\[ (T_{\mu \nu} - \frac{1}{2} g_{\mu \nu} g^{\lambda \sigma} T_{\lambda \sigma}) = \rho h^* u_\mu u_\nu + \left(\frac{1}{2} \rho h^* - p\right) g_{\mu \nu} - b_\mu b_\nu \]