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SpECTRE
v2026.04.01
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Computes the generalized harmonic upwind multipenalty boundary correction. More...
#include <UpwindPenalty.hpp>
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 | |
| 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 > | 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 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 |
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{align*} D_\beta = T_{\beta\hat{\beta}}^{\mathrm{ext}} \Lambda^{\mathrm{ext},-}_{\hat{\beta}\hat{\alpha}} v^{\mathrm{ext}}_{\hat{\alpha}} -T_{\beta\hat{\beta}}^{\mathrm{int}} \Lambda^{\mathrm{int},-}_{\hat{\beta}\hat{\alpha}} v^{\mathrm{int}}_{\hat{\alpha}}. \end{align*}
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 \(\Lambda_{\hat{\beta}\hat{\alpha}}^-\) 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 [200] 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{align*} D_{g_{ab}} &= \lambda_{v^{g}}^{\mathrm{ext},-} v^{\mathrm{ext},g}_{ab} - \lambda_{v^{g}}^{\mathrm{int},-} v^{\mathrm{int},g}_{ab}, \\ D_{\Pi_{ab}} &= \frac{1}{2}\left(\lambda_{v^+}^{\mathrm{ext},-} v^{\mathrm{ext},+}_{ab} + \lambda_{v^-}^{\mathrm{ext},-} v^{\mathrm{ext},-}_{ab}\right) + \lambda_{v^g}^{\mathrm{ext},-}\gamma_2 v^{\mathrm{ext},g}_{ab} \notag \\ &-\frac{1}{2}\left(\lambda_{v^+}^{\mathrm{int},-} v^{\mathrm{int},+}_{ab} + \lambda_{v^-}^{\mathrm{int},-} v^{\mathrm{int},-}_{ab}\right) - \lambda_{v^g}^{\mathrm{int},-}\gamma_2 v^{\mathrm{int},g}_{ab} , \\ D_{\Phi_{iab}} &= \frac{1}{2}\left(\lambda_{v^+}^{\mathrm{ext},-} v^{\mathrm{ext},+}_{ab} - \lambda_{v^-}^{\mathrm{ext},-} v^{\mathrm{ext},-}_{ab}\right)n_i^{\mathrm{ext}} + \lambda_{v^0}^{\mathrm{ext},-} v^{\mathrm{ext},0}_{iab} \notag \\ &- \frac{1}{2}\left(\lambda_{v^+}^{\mathrm{int},-} v^{\mathrm{int},+}_{ab} - \lambda_{v^-}^{\mathrm{int},-} v^{\mathrm{int},-}_{ab}\right)n_i^{\mathrm{int}} - \lambda_{v^0}^{\mathrm{int},-} v^{\mathrm{int},0}_{iab}, \end{align*}
with characteristic fields
\begin{align*} v^{g}_{ab} &= g_{ab}, \\ v^{0}_{iab} &= (\delta^k_i-n^k n_i)\Phi_{kab}, \\ v^{\pm}_{ab} &= \Pi_{ab}\pm n^i\Phi_{iab} -\gamma_2 g_{ab}, \end{align*}
and characteristic speeds
\begin{align*} \lambda_{v^g} =& -(1+\gamma_1) (\beta^i n_i + v^i_g n_i), \\ \lambda_{v^0} =& -\beta^i n_i -v^i_g n_i, \\ \lambda_{v^\pm} =& \pm \alpha - \beta^i n_i - v^i_g n_i, \end{align*}
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{align} \lambda^{\pm}_{\hat{\alpha}} = \left\{ \begin{array}{ll} \lambda_{\hat{\alpha}} & \mathrm{if}\;\pm\lambda_{\hat{\alpha}}> 0 \\ 0 & \mathrm{otherwise} \end{array}\right. \end{align}
In the implementation we store the speeds in a rank-4 tensor with the zeroth component being \(\lambda_{v^\Psi}\), the first being \(\lambda_{v^0}\), the second being \(\lambda_{v^+}\), and the third being \(\lambda_{v^-}\).
Note that we have assumed \(n_i^{\mathrm{ext}}\) points in the same direction as \(n_i^{\mathrm{int}}\), but in the code they point in opposite directions. If \(n_i^{\mathrm{ext}}\) points in the opposite direction the external speeds have their sign flipped and the \(\pm\) fields and their speeds reverse roles (i.e. the \(v^{\mathrm{ext},+}\) field is now flowing into the element, while \(v^{\mathrm{ext},-}\) flows out). In our implementation this reversal actually cancels out, and we have the following equations:
\begin{align*} D_{g_{ab}} &= -\lambda_{v^{g}}^{\mathrm{ext},+} v^{\mathrm{ext},g}_{ab} - \lambda_{v^{g}}^{\mathrm{int},-} v^{\mathrm{int},g}_{ab}, \\ D_{\Pi_{ab}} &= \frac{1}{2}\left(-\lambda_{v^+}^{\mathrm{ext},+} v^{\mathrm{ext},+}_{ab} - \lambda_{v^-}^{\mathrm{ext},+} v^{\mathrm{ext},-}_{ab}\right) - \lambda_{v^g}^{\mathrm{ext},+}\gamma_2 v^{\mathrm{ext},g}_{ab} \notag \\ &-\frac{1}{2}\left(\lambda_{v^+}^{\mathrm{int},-} v^{\mathrm{int},+}_{ab} + \lambda_{v^-}^{\mathrm{int},-} v^{\mathrm{int},-}_{ab}\right) - \lambda_{v^g}^{\mathrm{int},-}\gamma_2 v^{\mathrm{int},g}_{ab} , \\ D_{\Phi_{iab}} &= \frac{1}{2}\left(-\lambda_{v^+}^{\mathrm{ext},+} v^{\mathrm{ext},+}_{ab} + \lambda_{v^-}^{\mathrm{ext},+} v^{\mathrm{ext},-}_{ab}\right)n_i^{\mathrm{ext}} - \lambda_{v^0}^{\mathrm{ext},+} v^{\mathrm{ext},0}_{iab} \\ &- \frac{1}{2}\left(\lambda_{v^+}^{\mathrm{int},-} v^{\mathrm{int},+}_{ab} - \lambda_{v^-}^{\mathrm{int},-} v^{\mathrm{int},-}_{ab}\right)n_i^{\mathrm{int}} - \lambda_{v^0}^{\mathrm{int},-} v^{\mathrm{int},0}_{iab}, \end{align*}
| using gh::BoundaryCorrections::UpwindPenalty< Dim >::dg_package_data_temporary_tags |
| using gh::BoundaryCorrections::UpwindPenalty< Dim >::dg_package_field_tags |
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overridevirtual |
Implements evolution::BoundaryCorrection.
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staticconstexpr |