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SpECTRE
2021.08.02
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Items related to evolving a scalar wave on a curved background. More...
Namespaces | |
| namespace | AnalyticData |
| Holds classes implementing analytic data for the CurvedScalarWave system. | |
| namespace | BoundaryCorrections |
| Boundary corrections/numerical fluxes. | |
| namespace | Tags |
| Tags for the curved scalar wave system. | |
Classes | |
| struct | CharacteristicFieldsCompute |
| struct | CharacteristicSpeedsCompute |
| struct | ComputeNormalDotFluxes |
| Compute fluxes for the scalar-wave system in curved spacetime. More... | |
| struct | EvolvedFieldsFromCharacteristicFieldsCompute |
| struct | Phi |
| struct | Pi |
| struct | Psi |
| struct | System |
| struct | TimeDerivative |
| Compute the time derivative of the evolved variables of the first-order scalar wave system on a curved background. More... | |
Functions | |
| template<size_t SpatialDim> | |
| std::array< DataVector, 4 > | characteristic_speeds (const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| Compute the characteristic speeds for the scalar wave system in curved spacetime. More... | |
| template<size_t SpatialDim> | |
| void | characteristic_speeds (gsl::not_null< std::array< DataVector, 4 > * > char_speeds, const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| Compute the characteristic speeds for the scalar wave system in curved spacetime. More... | |
| template<size_t SpatialDim> | |
| void | characteristic_speeds (gsl::not_null< tnsr::a< DataVector, 3, Frame::Inertial > * > char_speeds, const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| Compute the characteristic speeds for the scalar wave system in curved spacetime. More... | |
| template<size_t SpatialDim> | |
| Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > > | characteristic_fields (const Scalar< DataVector > &gamma_2, const tnsr::II< DataVector, SpatialDim, Frame::Inertial > &inverse_spatial_metric, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| Computes characteristic fields from evolved fields. More... | |
| template<size_t SpatialDim> | |
| void | characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > > * > char_fields, const Scalar< DataVector > &gamma_2, const tnsr::II< DataVector, SpatialDim, Frame::Inertial > &inverse_spatial_metric, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| Computes characteristic fields from evolved fields. More... | |
| template<size_t SpatialDim> | |
| Variables< tmpl::list< Psi, Pi, Phi< SpatialDim > > > | evolved_fields_from_characteristic_fields (const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| For expressions used here to compute evolved fields from characteristic ones, see CharacteristicFieldsCompute. | |
| template<size_t SpatialDim> | |
| void | evolved_fields_from_characteristic_fields (gsl::not_null< Variables< tmpl::list< Psi, Pi, Phi< SpatialDim > > > * > evolved_fields, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form) noexcept |
| For expressions used here to compute evolved fields from characteristic ones, see CharacteristicFieldsCompute. | |
| template<size_t SpatialDim> | |
| tnsr::i< DataVector, SpatialDim, Frame::Inertial > | one_index_constraint (const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &d_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi) noexcept |
| Computes the scalar-wave one-index constraint. More... | |
| template<size_t SpatialDim> | |
| void | one_index_constraint (gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > constraint, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &d_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi) noexcept |
| Computes the scalar-wave one-index constraint. More... | |
| template<size_t SpatialDim> | |
| tnsr::ij< DataVector, SpatialDim, Frame::Inertial > | two_index_constraint (const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &d_phi) noexcept |
| Computes the scalar-wave 2-index constraint. More... | |
| template<size_t SpatialDim> | |
| void | two_index_constraint (gsl::not_null< tnsr::ij< DataVector, SpatialDim, Frame::Inertial > * > constraint, const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &d_phi) noexcept |
| Computes the scalar-wave 2-index constraint. More... | |
Items related to evolving a scalar wave on a curved background.
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noexcept |
Computes characteristic fields from evolved fields.
CharacteristicFieldsCompute and EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system in curved spacetime.
CharacteristicFieldsCompute computes characteristic fields as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [59] . Their names used here differ from this paper:
\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}
The characteristic fields \(u\) are given in terms of the evolved fields by Eq. (33) - (35) of [59], respectively:
\begin{align*} v^{\hat \psi} =& \psi \\ v^{\hat 0}_{i} =& (\delta^k_i - n_i n^k) \Phi_{k} := P^k_i \Phi_{k} \\ v^{\hat \pm} =& \Pi \pm n^i \Phi_{i} - \gamma_2\psi \end{align*}
where \(\psi\) is the scalar field, \(\Pi\) and \(\Phi_{i}\) are evolved fields introduced by first derivatives of \(\psi\), \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface.
EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(w\) in terms of the characteristic fields. This uses the inverse of above relations (c.f. Eq. (36) - (38) of [59] ):
\begin{align*} \psi =& v^{\hat \psi}, \\ \Pi =& \frac{1}{2}(v^{\hat +} + v^{\hat -}) + \gamma_2 v^{\hat \psi}, \\ \Phi_{i} =& \frac{1}{2}(v^{\hat +} - v^{\hat -}) n_i + v^{\hat 0}_{i}. \end{align*}
The corresponding characteristic speeds \(\lambda\) are computed by CharacteristicSpeedsCompute .
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noexcept |
Computes characteristic fields from evolved fields.
CharacteristicFieldsCompute and EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system in curved spacetime.
CharacteristicFieldsCompute computes characteristic fields as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [59] . Their names used here differ from this paper:
\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}
The characteristic fields \(u\) are given in terms of the evolved fields by Eq. (33) - (35) of [59], respectively:
\begin{align*} v^{\hat \psi} =& \psi \\ v^{\hat 0}_{i} =& (\delta^k_i - n_i n^k) \Phi_{k} := P^k_i \Phi_{k} \\ v^{\hat \pm} =& \Pi \pm n^i \Phi_{i} - \gamma_2\psi \end{align*}
where \(\psi\) is the scalar field, \(\Pi\) and \(\Phi_{i}\) are evolved fields introduced by first derivatives of \(\psi\), \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface.
EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(w\) in terms of the characteristic fields. This uses the inverse of above relations (c.f. Eq. (36) - (38) of [59] ):
\begin{align*} \psi =& v^{\hat \psi}, \\ \Pi =& \frac{1}{2}(v^{\hat +} + v^{\hat -}) + \gamma_2 v^{\hat \psi}, \\ \Phi_{i} =& \frac{1}{2}(v^{\hat +} - v^{\hat -}) n_i + v^{\hat 0}_{i}. \end{align*}
The corresponding characteristic speeds \(\lambda\) are computed by CharacteristicSpeedsCompute .
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noexcept |
Compute the characteristic speeds for the scalar wave system in curved spacetime.
Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [59] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:
\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}
The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [59] :
\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}
where \(n_k\) is the unit normal to the surface.
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noexcept |
Compute the characteristic speeds for the scalar wave system in curved spacetime.
Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [59] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:
\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}
The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [59] :
\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}
where \(n_k\) is the unit normal to the surface.
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noexcept |
Compute the characteristic speeds for the scalar wave system in curved spacetime.
Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [59] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:
\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}
The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [59] :
\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}
where \(n_k\) is the unit normal to the surface.
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noexcept |
Computes the scalar-wave one-index constraint.
Computes the scalar-wave one-index constraint, \(C_{i} = \partial_i\psi - \Phi_{i},\) which is given by Eq. (19) of [59]
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noexcept |
Computes the scalar-wave one-index constraint.
Computes the scalar-wave one-index constraint, \(C_{i} = \partial_i\psi - \Phi_{i},\) which is given by Eq. (19) of [59]
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noexcept |
Computes the scalar-wave 2-index constraint.
Computes the scalar-wave 2-index FOSH constraint [Eq. (20) of [59]],
\begin{eqnarray} C_{ij} &\equiv& \partial_i \Phi_j - \partial_j \Phi_i \end{eqnarray}
where \(\Phi_{i} = \partial_i\psi\); and \(\psi\) is the scalar field.
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noexcept |
Computes the scalar-wave 2-index constraint.
Computes the scalar-wave 2-index FOSH constraint [Eq. (20) of [59]],
\begin{eqnarray} C_{ij} &\equiv& \partial_i \Phi_j - \partial_j \Phi_i \end{eqnarray}
where \(\Phi_{i} = \partial_i\psi\); and \(\psi\) is the scalar field.