gh::BoundaryCorrections::AveragedUpwindPenalty< Dim > Class Template Reference

Computes the generalized harmonic upwind multipenalty boundary correction using the averaged fields across the interface. More...

#include <AveragedUpwindPenalty.hpp>

Classes
struct	MeshVelocity

Public Types
using	options = tmpl::list<>
using	dg_package_field_tags
using	dg_package_data_temporary_tags
using	dg_package_data_primitive_tags = tmpl::list<>
using	dg_package_data_volume_tags = tmpl::list<>
using	dg_boundary_terms_volume_tags = tmpl::list<>

Public Member Functions
	AveragedUpwindPenalty (const AveragedUpwindPenalty &)=default
AveragedUpwindPenalty &	operator= (const AveragedUpwindPenalty &)=default
	AveragedUpwindPenalty (AveragedUpwindPenalty &&)=default
AveragedUpwindPenalty &	operator= (AveragedUpwindPenalty &&)=default
void	pup (PUP::er &p) override
std::unique_ptr< BoundaryCorrection >	get_clone () const override
double	dg_package_data (gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_spacetime_metric, gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_pi, gsl::not_null< tnsr::iaa< DataVector, Dim, Frame::Inertial > * > packaged_phi, gsl::not_null< Scalar< DataVector > * > packaged_constraint_gamma1, gsl::not_null< Scalar< DataVector > * > packaged_constraint_gamma2, gsl::not_null< tnsr::i< DataVector, Dim, Frame::Inertial > * > packaged_normal, gsl::not_null< tnsr::I< DataVector, Dim, Frame::Inertial > * > packaged_mesh_velocity, const tnsr::aa< DataVector, Dim, Frame::Inertial > &spacetime_metric, const tnsr::aa< DataVector, Dim, Frame::Inertial > &pi, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &phi, const Scalar< DataVector > &constraint_gamma1, const Scalar< DataVector > &constraint_gamma2, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, Dim > &shift, const tnsr::i< DataVector, Dim, Frame::Inertial > &normal_covector, const tnsr::I< DataVector, Dim, Frame::Inertial > &normal_vector, const std::optional< tnsr::I< DataVector, Dim, Frame::Inertial > > &mesh_velocity, const std::optional< Scalar< DataVector > > &normal_dot_mesh_velocity) const
void	dg_boundary_terms (gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > boundary_correction_spacetime_metric, gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > boundary_correction_pi, gsl::not_null< tnsr::iaa< DataVector, Dim, Frame::Inertial > * > boundary_correction_phi, const tnsr::aa< DataVector, Dim, Frame::Inertial > &spacetime_metric_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &pi_int, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &phi_int, const Scalar< DataVector > &constraint_gamma1_int, const Scalar< DataVector > &constraint_gamma2_int, const tnsr::i< DataVector, Dim, Frame::Inertial > &normal_int, const tnsr::I< DataVector, Dim, Frame::Inertial > &mesh_velocity_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &spacetime_metric_ext, const tnsr::aa< DataVector, Dim, Frame::Inertial > &pi_ext, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &phi_ext, const Scalar< DataVector > &constraint_gamma1_ext, const Scalar< DataVector > &constraint_gamma2_ext, const tnsr::i< DataVector, Dim, Frame::Inertial > &normal_ext, const tnsr::I< DataVector, Dim, Frame::Inertial > &mesh_velocity_ext, dg::Formulation dg_formulation) const
Public Member Functions inherited from evolution::BoundaryCorrection
	BoundaryCorrection (const BoundaryCorrection &)=default
BoundaryCorrection &	operator= (const BoundaryCorrection &)=default
	BoundaryCorrection (BoundaryCorrection &&)=default
BoundaryCorrection &	operator= (BoundaryCorrection &&)=default
	BoundaryCorrection (CkMigrateMessage *msg)

Static Public Attributes
static constexpr Options::String	help

Detailed Description

template<size_t Dim>
class gh::BoundaryCorrections::AveragedUpwindPenalty< Dim >

Computes the generalized harmonic upwind multipenalty boundary correction using the averaged fields across the interface.

The correction is given by

\begin{equation} D_\beta = T^{-1}_{\beta\hat{\beta}}(u^{\text{avg}}) \Lambda^-_{\hat{\beta}\hat{\alpha}}(u^{\text{avg}}) T_{\hat{\alpha}\alpha}(u^{\text{avg}}) \Delta u_\alpha, \end{equation}

where \(u_\alpha\) are the evolved fields, \(\Lambda^-\) is the diagonal matrix of incoming characteristic speeds, \(T_{\hat{\alpha}\alpha}\) transforms from evolved to characteristic fields, and \(\Delta u_\alpha = u_\alpha^{\text{ext}} - u_\alpha^{\text{int}}\) is the discontinuity in the evolved fields. All factors except the boundary discontinuity are calculated using the averaged of the internal and external evolved fields:

\begin{equation} u^{\text{avg}} = \frac{u^{\text{int}} + u^{\text{ext}}}{2}. \end{equation}

For the first-order generalized harmonic system the correction is:

\begin{align} D_{g_{ab}} &= \lambda_{v^g}^- \Delta g_{ab} \\ D_{\Pi_{ab}} &= \left(\lambda_{v^g}^- - \frac{\lambda_{v^+}^- + \lambda_{v^-}^-}{2}\right) \gamma_2 \Delta g_{ab} + \frac{\lambda_{v^+}^- + \lambda_{v^-}^-}{2} \Delta \Pi_{ab} + \frac{\lambda_{v^+}^- - \lambda_{v^-}^-}{2} n^j \Delta \Phi_{jab} \\ D_{\Phi_{iab}} &= \lambda_{v^0}^- (\delta_i^j - n_i n^j) \Delta \Phi_{jab} + \frac{\lambda_{v^+}^- - \lambda_{v^-}^-}{2} n_i (\Delta \Pi_{ab} - \gamma_2 \Delta g_{ab}) + \frac{\lambda_{v^+}^- + \lambda_{v^-}^-}{2} n_i n^j \Delta \Phi_{jab}. \end{align}

In the above expressions, the outgoing face normal \(n_i\) is normalized using the average metric from the two sides. Quantities such as \(\gamma_2\) that should analytically be the same on both sides of the interface are also averaged, as they can differ because of truncation errors during projection.

The characteristic speeds are given by

\begin{align} \lambda_{v^g} &= - (1 + \gamma_1) (\beta^i + v^i_g) n_i, \\ \lambda_{v^0} &= - (\beta^i + v^i_g) n_i, \\ \lambda_{v^\pm} &= \pm \alpha - (\beta^i + v^i_g) n_i, \end{align}

where \(v_g^i\) is the mesh velocity, with the incoming speeds

\begin{equation} \lambda^-_{\hat{\alpha}} = \begin{cases} \lambda_{\hat{\alpha}} & \lambda_{\hat{\alpha}} < 0 \\ 0 & \text{otherwise.} \end{cases} \end{equation}

Member Typedef Documentation

dg_package_data_temporary_tags

template<size_t Dim>

using gh::BoundaryCorrections::AveragedUpwindPenalty< Dim >::dg_package_data_temporary_tags

Initial value:

 
      tmpl::list<gh::Tags::ConstraintGamma1, gh::Tags::ConstraintGamma2,
                 gr::Tags::Lapse<DataVector>, gr::Tags::Shift<DataVector, Dim>>

dg_package_field_tags

template<size_t Dim>

using gh::BoundaryCorrections::AveragedUpwindPenalty< Dim >::dg_package_field_tags

Initial value:

 
      tmpl::list<gr::Tags::SpacetimeMetric<DataVector, Dim>,
                 gh::Tags::Pi<DataVector, Dim>, gh::Tags::Phi<DataVector, Dim>,
                 gh::Tags::ConstraintGamma1, gh::Tags::ConstraintGamma2,
                 domain::Tags::UnnormalizedFaceNormal<Dim>, MeshVelocity>

Member Function Documentation

get_clone()

template<size_t Dim>

std::unique_ptr< BoundaryCorrection > gh::BoundaryCorrections::AveragedUpwindPenalty< Dim >::get_clone ( ) const

overridevirtual

Implements evolution::BoundaryCorrection.

Member Data Documentation

help

template<size_t Dim>

Options::String gh::BoundaryCorrections::AveragedUpwindPenalty< Dim >::help

staticconstexpr

Initial value:

= {
      "Computes the upwind penalty boundary correction term for the "
      "generalized harmonic system using the averaged fields from the two "
      "sides of the interface."}

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The documentation for this class was generated from the following file:

• src/Evolution/Systems/GeneralizedHarmonic/BoundaryCorrections/AveragedUpwindPenalty.hpp