Compute the HLLC numerical flux. More...

#include <Hllc.hpp>

Classes
struct	LargestIngoingSpeed
	Estimate for one of the signal speeds. More...

struct	LargestOutgoingSpeed
	Estimate for the other signal speed. More...

struct	NormalVelocity
	The normal component of the velocity. More...

Public Types
using	char_speeds_tag = Tags::CharacteristicSpeeds< Dim >

using	options = tmpl::list<>

using	variables_tags = tmpl::list< Tags::MassDensityCons, Tags::MomentumDensity< Dim >, Tags::EnergyDensity >

using	package_field_tags = tmpl::list< ::Tags::NormalDotFlux< Tags::MassDensityCons >, ::Tags::NormalDotFlux< Tags::MomentumDensity< Dim, Frame > >, ::Tags::NormalDotFlux< Tags::EnergyDensity >, Tags::MassDensityCons, Tags::MomentumDensity< Dim, Frame >, Tags::EnergyDensity, Tags::Pressure< DataVector >, ::Tags::Normalized< domain::Tags::UnnormalizedFaceNormal< Dim, Frame > >, NormalVelocity, LargestIngoingSpeed, LargestOutgoingSpeed >

using	package_extra_tags = tmpl::list<>

using	argument_tags = tmpl::list< ::Tags::NormalDotFlux< Tags::MassDensityCons >, ::Tags::NormalDotFlux< Tags::MomentumDensity< Dim, Frame > >, ::Tags::NormalDotFlux< Tags::EnergyDensity >, Tags::MassDensityCons, Tags::MomentumDensity< Dim, Frame >, Tags::EnergyDensity, Tags::Velocity< DataVector, Dim, Frame >, Tags::Pressure< DataVector >, char_speeds_tag, ::Tags::Normalized< domain::Tags::UnnormalizedFaceNormal< Dim, Frame > > >

Public Member Functions
void	pup (PUP::er &) noexcept

void	package_data (gsl::not_null< Scalar< DataVector > * > packaged_n_dot_f_mass_density, gsl::not_null< tnsr::I< DataVector, Dim, Frame > * > packaged_n_dot_f_momentum_density, gsl::not_null< Scalar< DataVector > * > packaged_n_dot_f_energy_density, gsl::not_null< Scalar< DataVector > * > packaged_mass_density, gsl::not_null< tnsr::I< DataVector, Dim, Frame > * > packaged_momentum_density, gsl::not_null< Scalar< DataVector > * > packaged_energy_density, gsl::not_null< Scalar< DataVector > * > packaged_pressure, gsl::not_null< tnsr::i< DataVector, Dim, Frame > * > packaged_face_normal, gsl::not_null< Scalar< DataVector > * > packaged_normal_velocity, gsl::not_null< Scalar< DataVector > * > packaged_largest_ingoing_speed, gsl::not_null< Scalar< DataVector > * > packaged_largest_outgoing_speed, const Scalar< DataVector > &normal_dot_flux_mass_density, const tnsr::I< DataVector, Dim, Frame > &normal_dot_flux_momentum_density, const Scalar< DataVector > &normal_dot_flux_energy_density, const Scalar< DataVector > &mass_density, const tnsr::I< DataVector, Dim, Frame > &momentum_density, const Scalar< DataVector > &energy_density, const tnsr::I< DataVector, Dim, Frame > &velocity, const Scalar< DataVector > &pressure, const typename char_speeds_tag::type &characteristic_speeds, const tnsr::i< DataVector, Dim, Frame > &interface_unit_normal) const noexcept

void	operator() (gsl::not_null< Scalar< DataVector > * > normal_dot_numerical_flux_mass_density, gsl::not_null< tnsr::I< DataVector, Dim, Frame > * > normal_dot_numerical_flux_momentum_density, gsl::not_null< Scalar< DataVector > * > normal_dot_numerical_flux_energy_density, const Scalar< DataVector > &normal_dot_flux_mass_density_int, const tnsr::I< DataVector, Dim, Frame > &normal_dot_flux_momentum_density_int, const Scalar< DataVector > &normal_dot_flux_energy_density_int, const Scalar< DataVector > &mass_density_int, const tnsr::I< DataVector, Dim, Frame > &momentum_density_int, const Scalar< DataVector > &energy_density_int, const Scalar< DataVector > &pressure_int, const tnsr::i< DataVector, Dim, Frame > &interface_unit_normal, const Scalar< DataVector > &normal_velocity_int, const Scalar< DataVector > &largest_ingoing_speed_int, const Scalar< DataVector > &largest_outgoing_speed_int, const Scalar< DataVector > &minus_normal_dot_flux_mass_density_ext, const tnsr::I< DataVector, Dim, Frame > &minus_normal_dot_flux_momentum_density_ext, const Scalar< DataVector > &minus_normal_dot_flux_energy_density_ext, const Scalar< DataVector > &mass_density_ext, const tnsr::I< DataVector, Dim, Frame > &momentum_density_ext, const Scalar< DataVector > &energy_density_ext, const Scalar< DataVector > &pressure_ext, const tnsr::i< DataVector, Dim, Frame > &minus_interface_unit_normal, const Scalar< DataVector > &minus_normal_velocity_ext, const Scalar< DataVector > &minus_largest_outgoing_speed_ext, const Scalar< DataVector > &minus_largest_ingoing_speed_ext) const noexcept

Static Public Attributes
static constexpr Options::String	help

Detailed Description

template<size_t Dim, typename Frame>
struct NewtonianEuler::NumericalFluxes::Hllc< Dim, Frame >

Compute the HLLC numerical flux.

Class implementing the HLLC flux for the Newtonian Euler equations, originally introduced by E. F. Toro, M. Spruce and W. Speares [107]. Let \(F^k = [F^k(\rho), F^k(S^i), F^k(e)]^T\) be the fluxes of the conservative quantities \(U = [\rho, S^i, e]^T\) (see NewtonianEuler::ComputeFluxes). Denoting \(v_n = n_k v^k\) and \(F = n_kF^k\), where \(v^k\) is the velocity and \(n_k\) is the interface unit normal, the HLLC flux is

\begin{align*} G_\text{HLLC} = \begin{cases} F_\text{int}, & S_\text{min} > 0, \\ G_{*\text{int}}, & S_\text{min} \leq 0\quad\text{and}\quad S_* > 0, \\ G_{*\text{ext}}, & S_* \leq 0\quad\text{and}\quad S_\text{max} > 0, \\ F_\text{ext}, & S_\text{max} \leq 0, \end{cases} \end{align*}

where

\begin{align*} G_{*\text{int}} &= \frac{S_*\left(F_\text{int} - S_\text{min}U_\text{int}\right) - S_\text{min} p_{*\text{int}}D_*}{S_* - S_\text{min}},\\ G_{*\text{ext}} &= \frac{S_*\left(F_\text{ext} - S_\text{max}U_\text{ext}\right) - S_\text{max} p_{*\text{ext}}D_*}{S_* - S_\text{max}}, \end{align*}

with

\begin{align*} p_{*\text{int}} &\equiv p_\text{int} + \rho_\text{int} \left[(v_n)_\text{int} - S_\text{min}\right] \left[(v_n)_\text{int} - S_*\right], \\ p_{*\text{ext}} &\equiv p_\text{ext} + \rho_\text{ext} \left[(v_n)_\text{ext} - S_\text{max}\right] \left[(v_n)_\text{ext} - S_*\right], \\ S_* &\equiv \frac{p_\text{int} + F(\rho)_\text{int}\left[(v_n)_\text{int} - S_\text{min}\right] - p_\text{ext} - F(\rho)_\text{ext}\left[(v_n)_\text{ext} - S_\text{max}\right]}{F(\rho)_\text{int} - \rho_\text{int}S_\text{min} - F(\rho)_\text{ext} + \rho_\text{ext}S_\text{max}},\\ D_* &\equiv \left[\begin{array}{c} 0\\ n^i\\ S_* \end{array}\right], \end{align*}

and \(S_\text{min}\) and \(S_\text{max}\) are estimates of the minimum and maximum signal speeds bounding the ingoing and outgoing wavespeeds that arise when solving the Riemann problem. One requires \(S_\text{min} \leq S_* \leq S_\text{max}\). As estimates, we use

\begin{align*} S_\text{min} &= \text{min}\left(\{\lambda_\text{int}\},\{\lambda_\text{ext}\}, 0\right)\\ S_\text{max} &= \text{max}\left(\{\lambda_\text{int}\},\{\lambda_\text{ext}\}, 0\right), \end{align*}

where \(\{\lambda\}\) is the set of all the characteristic speeds along a given normal. This way, the definition of \(G_\text{HLLC}\) simplifies to

\begin{align*} G_\text{HLLC} = \begin{cases} \dfrac{S_*\left(F_\text{int} - S_\text{min}U_\text{int}\right) - S_\text{min} p_{*\text{int}}D_*}{S_* - S_\text{min}}, & S_* > 0, \\ \dfrac{S_*\left(F_\text{ext} - S_\text{max}U_\text{ext}\right) - S_\text{max} p_{*\text{ext}}D_*}{S_* - S_\text{max}}, & S_* \leq 0. \\ \end{cases} \end{align*}

Warning: The HLLC flux implemented here does not incorporate any cure for the Carbuncle phenomenon or other shock instabilities reported in the literature. Prefer using another numerical flux in more than 1-d.

Member Data Documentation

◆ help

template<size_t Dim, typename Frame >

constexpr Options::String NewtonianEuler::NumericalFluxes::Hllc< Dim, Frame >::help

staticconstexpr

Initial value:

= {

"Compute the HLLC flux for the Newtonian Euler system."}

Classes

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Member Data Documentation

◆ help