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SpECTRE
v2024.04.12
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Compute the primitive variables from the conservative variables. More...
#include <PrimitiveFromConservative.hpp>
Public Types | |
| using | return_tags = tmpl::list< hydro::Tags::RestMassDensity< DataVector >, hydro::Tags::SpecificInternalEnergy< DataVector >, hydro::Tags::LorentzFactor< DataVector >, hydro::Tags::SpecificEnthalpy< DataVector >, hydro::Tags::Pressure< DataVector >, hydro::Tags::SpatialVelocity< DataVector, Dim > > |
| using | argument_tags = tmpl::list< RelativisticEuler::Valencia::Tags::TildeD, RelativisticEuler::Valencia::Tags::TildeTau, RelativisticEuler::Valencia::Tags::TildeS< Dim >, gr::Tags::InverseSpatialMetric< DataVector, Dim >, gr::Tags::SqrtDetSpatialMetric< DataVector >, hydro::Tags::EquationOfStateBase > |
Static Public Member Functions | |
| template<size_t ThermodynamicDim> | |
| static void | apply (gsl::not_null< Scalar< DataVector > * > rest_mass_density, gsl::not_null< Scalar< DataVector > * > specific_internal_energy, gsl::not_null< Scalar< DataVector > * > lorentz_factor, gsl::not_null< Scalar< DataVector > * > specific_enthalpy, gsl::not_null< Scalar< DataVector > * > pressure, gsl::not_null< tnsr::I< DataVector, Dim, Frame::Inertial > * > spatial_velocity, const Scalar< DataVector > &tilde_d, const Scalar< DataVector > &tilde_tau, const tnsr::i< DataVector, Dim, Frame::Inertial > &tilde_s, const tnsr::II< DataVector, Dim, Frame::Inertial > &inv_spatial_metric, const Scalar< DataVector > &sqrt_det_spatial_metric, const EquationsOfState::EquationOfState< true, ThermodynamicDim > &equation_of_state) |
Compute the primitive variables from the conservative variables.
For the Valencia formulation of the Relativistic Euler system, the conversion of the evolved conserved variables to the primitive variables cannot be expressed in closed analytic form and requires a root find. We use the method from [Appendix C of Galeazzi et al, PRD 88, 064009 (2013)] (https://journals.aps.org/prd/abstract/10.1103/PhysRevD.88.064009). The method finds the root of
\[ f(z) = z - \frac{r}{h(z)} \]
where
\begin{align*} r = & \frac{\sqrt{\gamma^{mn} {\tilde S}_m {\tilde S}_n}}{\tilde D} \\ h(z) = & [1+ \epsilon(z)][1 + a(z)] \\ \epsilon(z) = & W(z) q - z r + \frac{z^2}{1 + W(z)} \\ W(z) = & \sqrt{1 + z^2} \\ q = & \frac{\tilde \tau}{\tilde D} \\ a(z) = & \frac{p}{\rho(z) [1 + \epsilon(z)]} \\ \rho(z) = & \frac{\tilde D}{\sqrt{\gamma} W(z)} \end{align*}
and where the conserved variables \({\tilde D}\), \({\tilde S}_i\), and \({\tilde \tau}\) are a generalized mass-energy density, momentum density, and specific internal energy density as measured by an Eulerian observer, \(\gamma\) and \(\gamma^{mn}\) are the determinant and inverse of the spatial metric \(\gamma_{mn}\), \(\rho\) is the rest mass density, \(W = 1/\sqrt{1 - \gamma_{mn} v^m v^n}\) 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, and \(p\) is the pressure. The pressure is determined from the equation of state. Finally, once \(z\) is found, the spatial velocity is given by
\[ v^i = \frac{{\tilde S}^i}{{\tilde D} W(z) h(z)} \]