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

Computes the generalized harmonic upwind multipenalty boundary correction. More...

#include <UpwindPenalty.hpp>

Public Types
using	options = implementation defined
using	dg_package_field_tags = implementation defined
using	dg_package_data_temporary_tags = implementation defined
using	dg_package_data_primitive_tags = implementation defined
using	dg_package_data_volume_tags = implementation defined
using	dg_boundary_terms_volume_tags = implementation defined
Public Types inherited from gh::BoundaryCorrections::BoundaryCorrection< Dim >
using	creatable_classes = implementation defined

Public Member Functions
	UpwindPenalty (const UpwindPenalty &)=default
UpwindPenalty &	operator= (const UpwindPenalty &)=default
	UpwindPenalty (UpwindPenalty &&)=default
UpwindPenalty &	operator= (UpwindPenalty &&)=default
void	pup (PUP::er &p) override
std::unique_ptr< BoundaryCorrection< Dim > >	get_clone () const override
double	dg_package_data (gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_v_spacetime_metric, gsl::not_null< tnsr::iaa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_v_zero, gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_v_plus, gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_v_minus, gsl::not_null< tnsr::iaa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_n_times_v_plus, gsl::not_null< tnsr::iaa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_n_times_v_minus, gsl::not_null< tnsr::aa< DataVector, Dim, Frame::Inertial > * > packaged_char_speed_gamma2_v_spacetime_metric, gsl::not_null< tnsr::a< DataVector, 3, Frame::Inertial > * > packaged_char_speeds, 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, Frame::Inertial > &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 > > &, 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 > &char_speed_v_spacetime_metric_int, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_v_zero_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_v_plus_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_v_minus_int, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_normal_times_v_plus_int, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_normal_times_v_minus_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_constraint_gamma2_v_spacetime_metric_int, const tnsr::a< DataVector, 3, Frame::Inertial > &char_speeds_int, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_v_spacetime_metric_ext, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_v_zero_ext, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_v_plus_ext, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_v_minus_ext, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_normal_times_v_plus_ext, const tnsr::iaa< DataVector, Dim, Frame::Inertial > &char_speed_normal_times_v_minus_ext, const tnsr::aa< DataVector, Dim, Frame::Inertial > &char_speed_constraint_gamma2_v_spacetime_metric_ext, const tnsr::a< DataVector, 3, Frame::Inertial > &char_speeds_ext, dg::Formulation) const
Public Member Functions inherited from gh::BoundaryCorrections::BoundaryCorrection< Dim >
	BoundaryCorrection (const BoundaryCorrection &)=default
BoundaryCorrection &	operator= (const BoundaryCorrection &)=default
	BoundaryCorrection (BoundaryCorrection &&)=default
BoundaryCorrection &	operator= (BoundaryCorrection &&)=default
virtual std::unique_ptr< BoundaryCorrection< Dim > >	get_clone () const =0

Static Public Attributes
static constexpr Options::String	help

Detailed Description

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

Computes the generalized harmonic upwind multipenalty boundary correction.

This implements the upwind multipenalty boundary correction term $D_{\beta }$. The general form is given by:

$$\begin{array}{r}{D}_{\beta }=T_{\beta \hat{\beta }}^{\mathrm{e}\mathrm{x}\mathrm{t}}{\mathrm{\Lambda }}_{\hat{\beta }\hat{\alpha }}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{\hat{\alpha }}^{\mathrm{e}\mathrm{x}\mathrm{t}}-T_{\beta \hat{\beta }}^{\mathrm{i}\mathrm{n}\mathrm{t}}{\mathrm{\Lambda }}_{\hat{\beta }\hat{\alpha }}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{\hat{\alpha }}^{\mathrm{i}\mathrm{n}\mathrm{t}}.\end{array}$$

We denote the evolved fields by $u_{\alpha }$, the characteristic fields by $v_{\hat{\alpha }}$, and implicitly sum over reapeated indices. $T_{\beta \hat{\beta }}$ transforms characteristic fields to evolved fields, while ${\mathrm{\Lambda }}_{\hat{\beta }\hat{\alpha }}^{-}$ is a diagonal matrix with only the negative characteristic speeds, and has zeros on the diagonal for all other entries. The int and ext superscripts denote quantities on the internal and external side of the mortar. Note that Eq. (6.3) of [192] is not exactly what's implemented since that boundary term does not consistently treat both sides of the interface on the same footing.

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

$$\begin{array}{rl}{D}_{g_{ab}}& ={\lambda }_{v^g}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},g}-{\lambda }_{v^g}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},g},\\ D_{{\mathrm{\Pi }}_{ab}}& =\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},+}+{\lambda }_{v^{-}}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},-})+{\lambda }_{v^g}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{\gamma }_2{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},g}\\ & -\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},+}+{\lambda }_{v^{-}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},-})-{\lambda }_{v^g}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{\gamma }_2{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},g},\\ D_{{\mathrm{\Phi }}_{iab}}& =\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},+}-{\lambda }_{v^{-}}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},-})n_i^{\mathrm{e}\mathrm{x}\mathrm{t}}+{\lambda }_{v^0}^{\mathrm{e}\mathrm{x}\mathrm{t},-}{v}_{iab}^{\mathrm{e}\mathrm{x}\mathrm{t},0}\\ & -\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},+}-{\lambda }_{v^{-}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},-})n_i^{\mathrm{i}\mathrm{n}\mathrm{t}}-{\lambda }_{v^0}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{iab}^{\mathrm{i}\mathrm{n}\mathrm{t},0},\end{array}$$

with characteristic fields

$$\begin{array}{rl}{v}_{ab}^g& =g_{ab},\\ v_{iab}^0& =({\delta }_i^k-n^k{n}_i){\mathrm{\Phi }}_{kab},\\ v_{ab}^{\pm }& ={\mathrm{\Pi }}_{ab}\pm n^i{\mathrm{\Phi }}_{iab}-{\gamma }_2{g}_{ab},\end{array}$$

and characteristic speeds

$$\begin{array}{rl}{\lambda }_{v^g}=& -(1+{\gamma }_1){\beta }^i{n}_i-v_g^i{n}_i,\\ {\lambda }_{v^0}=& -{\beta }^i{n}_i-v_g^i{n}_i,\\ {\lambda }_{v^{\pm }}=& \pm \alpha -{\beta }^i{n}_i-v_g^i{n}_i,\end{array}$$

where $v_g^i$ is the mesh velocity and $n_i$ is the outward directed unit normal covector to the interface. We have also defined

$$\begin{array}{}\text{(1)}& {\lambda }_{\hat{\alpha }}^{\pm }=\{\begin{array}{ll}{\lambda }_{\hat{\alpha }}& \mathrm{i}\mathrm{f}\pm {\lambda }_{\hat{\alpha }}>0\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\end{array}$$

In the implementation we store the speeds in a rank-4 tensor with the zeroth component being ${\lambda }_{v^{\mathrm{\Psi }}}$, the first being ${\lambda }_{v^0}$, the second being ${\lambda }_{v^{+}}$, and the third being ${\lambda }_{v^{-}}$.

Note that we have assumed $n_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ points in the same direction as $n_i^{\mathrm{i}\mathrm{n}\mathrm{t}}$, but in the code they point in opposite directions. If $n_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ points in the opposite direction the external speeds have their sign flipped and the $\pm$ fields and their speeds reverse roles (i.e. the $v^{\mathrm{e}\mathrm{x}\mathrm{t},+}$ field is now flowing into the element, while $v^{\mathrm{e}\mathrm{x}\mathrm{t},-}$ flows out). In our implementation this reversal actually cancels out, and we have the following equations:

$$\begin{array}{rl}{D}_{g_{ab}}& =-{\lambda }_{v^g}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},g}-{\lambda }_{v^g}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},g},\\ D_{{\mathrm{\Pi }}_{ab}}& =\frac{1}{2}(-{\lambda }_{v^{+}}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},+}-{\lambda }_{v^{-}}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},-})-{\lambda }_{v^g}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{\gamma }_2{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},g}\\ & -\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},+}+{\lambda }_{v^{-}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},-})-{\lambda }_{v^g}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{\gamma }_2{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},g},\\ D_{{\mathrm{\Phi }}_{iab}}& =\frac{1}{2}(-{\lambda }_{v^{+}}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},+}+{\lambda }_{v^{-}}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{ab}^{\mathrm{e}\mathrm{x}\mathrm{t},-})n_i^{\mathrm{e}\mathrm{x}\mathrm{t}}-{\lambda }_{v^0}^{\mathrm{e}\mathrm{x}\mathrm{t},+}{v}_{iab}^{\mathrm{e}\mathrm{x}\mathrm{t},0}\\ & -\frac{1}{2}({\lambda }_{v^{+}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},+}-{\lambda }_{v^{-}}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{ab}^{\mathrm{i}\mathrm{n}\mathrm{t},-})n_i^{\mathrm{i}\mathrm{n}\mathrm{t}}-{\lambda }_{v^0}^{\mathrm{i}\mathrm{n}\mathrm{t},-}{v}_{iab}^{\mathrm{i}\mathrm{n}\mathrm{t},0},\end{array}$$

Member Function Documentation

get_clone()

template<size_t Dim>

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

overridevirtual

Implements gh::BoundaryCorrections::BoundaryCorrection< Dim >.

Member Data Documentation

help

template<size_t Dim>

constexpr Options::String gh::BoundaryCorrections::UpwindPenalty< Dim >::help

staticconstexpr

Initial value:

= {
      "Computes the UpwindPenalty boundary correction term for the generalized "
      "harmonic system."}

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

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