Implements constraint-preserving boundary conditions with a second order Bayliss-Turkel radiation boundary condition. More...
#include <ConstraintPreservingSphericalRadiation.hpp>
Public Types | |
| using | options = tmpl::list<> |
| using | dg_interior_evolved_variables_tags = tmpl::list< Tags::Psi, Tags::Phi< Dim > > |
| using | dg_interior_temporary_tags = tmpl::list< domain::Tags::Coordinates< Dim, Frame::Inertial >, Tags::ConstraintGamma1, Tags::ConstraintGamma2, gr::Tags::Lapse< DataVector >, gr::Tags::Shift< DataVector, Dim > > |
| using | dg_interior_dt_vars_tags = tmpl::list<::Tags::dt< Tags::Psi >, ::Tags::dt< Tags::Pi >, ::Tags::dt< Tags::Phi< Dim > > > |
| using | dg_interior_deriv_vars_tags = tmpl::list< ::Tags::deriv< Tags::Psi, tmpl::size_t< Dim >, Frame::Inertial >, ::Tags::deriv< Tags::Pi, tmpl::size_t< Dim >, Frame::Inertial >, ::Tags::deriv< Tags::Phi< Dim >, tmpl::size_t< Dim >, Frame::Inertial > > |
| using | dg_gridless_tags = tmpl::list<> |
Public Member Functions | |
| ConstraintPreservingSphericalRadiation (ConstraintPreservingSphericalRadiation &&)=default | |
| ConstraintPreservingSphericalRadiation & | operator= (ConstraintPreservingSphericalRadiation &&)=default |
| ConstraintPreservingSphericalRadiation (const ConstraintPreservingSphericalRadiation &)=default | |
| ConstraintPreservingSphericalRadiation & | operator= (const ConstraintPreservingSphericalRadiation &)=default |
| ConstraintPreservingSphericalRadiation (CkMigrateMessage *msg) | |
| WRAPPED_PUPable_decl_base_template (domain::BoundaryConditions::BoundaryCondition, ConstraintPreservingSphericalRadiation) | |
| auto | get_clone () const -> std::unique_ptr< domain::BoundaryConditions::BoundaryCondition > override |
| void | pup (PUP::er &p) override |
| std::optional< std::string > | dg_time_derivative (gsl::not_null< Scalar< DataVector > * > dt_psi_correction, gsl::not_null< Scalar< DataVector > * > dt_pi_correction, gsl::not_null< tnsr::i< DataVector, Dim, Frame::Inertial > * > dt_phi_correction, const std::optional< tnsr::I< DataVector, Dim > > &face_mesh_velocity, const tnsr::i< DataVector, Dim > &normal_covector, const tnsr::I< DataVector, Dim > &normal_vector, const Scalar< DataVector > &psi, const tnsr::i< DataVector, Dim > &phi, const tnsr::I< DataVector, Dim, Frame::Inertial > &coords, const Scalar< DataVector > &gamma1, const Scalar< DataVector > &gamma2, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, Dim > &shift, const Scalar< DataVector > &logical_dt_psi, const Scalar< DataVector > &logical_dt_pi, const tnsr::i< DataVector, Dim > &logical_dt_phi, const tnsr::i< DataVector, Dim > &d_psi, const tnsr::i< DataVector, Dim > &d_pi, const tnsr::ij< DataVector, Dim > &d_phi) const |
 Public Member Functions inherited from CurvedScalarWave::BoundaryConditions::BoundaryCondition< Dim > | |
| BoundaryCondition (BoundaryCondition &&)=default | |
| BoundaryCondition & | operator= (BoundaryCondition &&)=default |
| BoundaryCondition (const BoundaryCondition &)=default | |
| BoundaryCondition & | operator= (const BoundaryCondition &)=default |
| BoundaryCondition (CkMigrateMessage *msg) | |
| void | pup (PUP::er &p) override |
| Public Member Functions inherited from domain::BoundaryConditions::BoundaryCondition | |
| BoundaryCondition (BoundaryCondition &&)=default | |
| BoundaryCondition & | operator= (BoundaryCondition &&)=default |
| BoundaryCondition (const BoundaryCondition &)=default | |
| BoundaryCondition & | operator= (const BoundaryCondition &)=default |
| BoundaryCondition (CkMigrateMessage *const msg) | |
| WRAPPED_PUPable_abstract (BoundaryCondition) | |
| virtual auto | get_clone () const -> std::unique_ptr< BoundaryCondition >=0 |
Static Public Attributes | |
| static constexpr Options::String | help |
| static constexpr evolution::BoundaryConditions::Type | bc_type |
Detailed Description
class CurvedScalarWave::BoundaryConditions::ConstraintPreservingSphericalRadiation< Dim >
Implements constraint-preserving boundary conditions with a second order Bayliss-Turkel radiation boundary condition.
Details
The Bayliss-Turkel boundary conditions are technically only valid in flat space and should therefore only be used at boundaries where the background spacetime is approximately Minkwoski such as (sufficiently far out) outer boundaries for asymptotically flat spacetimes. Small reflections are still likely to occur.
The constraint-preserving part of the boundary conditions are set on the time derivatives of all evolved fields. The physical Bayliss-Turkel boundary conditions are additionally set onto the time derivative of \(\Pi\).
The constraints are defined as follows:
\begin{align*} \mathcal{C}_i&=\partial_i\Psi - \Phi_i=0, \\ \mathcal{C}_{ij}&=\partial_{[i}\Phi_{j]}=0 \end{align*}
Inspection of the constraint evolution system (Eqs. 29-30 in [89]) shows that the constraints themselves are characteristic fields. We can derive constraint boundary conditions the same way [102] does for the Einstein equations:
We express the constraints in terms of (evolution) characeristic fields and demand that the normal component of the constraint has to be zero when flowing into the boundary i.e. there are no constraints flowing into our numerical domain:
\begin{align*} 0 &= n^i \mathcal{C}_i &= n^i \partial_i w^\Psi - \frac{1}{2}(w^{+} - w^-) + n^i w_i^0 \\ (n^i \partial_i w^\Psi)_{BC} &= \frac{1}{2}(w^{+} - w^-) - n^i w_i^0 \end{align*}
and
\begin{align*} 0 &= 2 n^i \mathcal{C}_{ij} = n^i \partial_i w^0_j + \frac{1}{2}n^i n_j (\partial_i w^+ - \partial_i w^-) - \frac{1}{2}(\partial_j w^+ - \partial_j w^-) - n^i \partial_j w^0_i \\ (n^i \partial_i w^0_j)_{BC} &= - \frac{1}{2}n^i n_j (\partial_i w^+ - \partial_i w^-) + \frac{1}{2}(\partial_j w^+ + \partial_j w^-) + n^i \partial_j w^0_i \end{align*}
This condition is applied to the time derivative using the Bjorhus condition [22] :
\begin{align*} \partial_t u^\alpha + A^{i \alpha}_\beta \partial_i u^\beta &= F^\alpha \\ e^{\hat{\alpha}}_\alpha (\partial_t u^\alpha + A^{i \alpha}_\beta \partial_i u^\beta) &= e^{\hat{\alpha}}_\alpha F^\alpha \\ d_t u^{\hat{\alpha}} + e^{\hat{\alpha}}_\alpha A^{i \alpha}_\beta(P^k_i + n^k n_i) \partial_k u^\beta &= e^{\hat{\alpha}}_\alpha F^\alpha \\ d_t u^{\hat{\alpha}} + \lambda_{(\hat{\alpha})} n^k \partial_k u^{\hat{\alpha}} + e^{\hat{\alpha}}_\alpha A^{i \alpha}_\beta P^k_i \partial_k u^\beta &= e^{\hat{\alpha}}_\alpha F^\alpha \end{align*}
Defining the volume time derivative of the characteristic fields as:
\begin{equation*} D_t u^{\hat{\alpha}} \equiv e^{\hat{\alpha}}_\alpha (- A^{i \alpha}_\beta \partial_i u^\beta + F^\alpha) \end{equation*}
The boundary conditions are now formulated as follows:
\begin{equation*} d_t u^{\hat{\alpha}} = D_t u^{\hat{\alpha}} + \lambda_{(\hat{\alpha})} (n^i\partial_i u^{\hat{\alpha}} - (n^i\partial_i u^{\hat{\alpha}})_{BC}) \end{equation*}
Using the condition that there are no incoming constraint fields, this gives:
\begin{align*} d_t Z^1 &= D_t w^\Psi + \lambda_\Psi n^i \mathcal{C}_i \\ d_t Z^2_i &= D_t w^0_i + 2 \lambda_0 n^i \mathcal{C}_{ij} \end{align*}
The Bayliss-Turkel boundary conditions are given by:
\begin{align*} \prod_{l=1}^m\left(\partial_t + \partial_r + \frac{2l-1}{r}\right)\Psi=0, \end{align*}
which we expand here to second order ( \(m=2\)) to derive conditions for \(\partial_t\Pi^{\mathrm{BC}}\):
\begin{align*} \partial_t\Pi^{\mathrm{BC}} &=\left(\partial_t\partial_r + \partial_r\partial_t + \partial_r^2+\frac{4}{r}\partial_t +\frac{4}{r}\partial_r + \frac{2}{r^2}\right)\Psi \notag \\ &=\left((2n^i + \beta^i) \partial_t \Phi_i + n^i n^j\partial_i\Phi_j + \frac{4}{r}\partial_t\Psi + \frac{4}{r}n^i\Phi_i + \frac{2}{r^2}\Psi \right) / \alpha. \end{align*}
This derivation makes the following assumptions:
- The lapse, shift, normal vector and radius are time-independent, \(\partial_t \alpha = \partial_t \beta^i = \partial_t n^i = \partial_t r = 0\). If necessary, these time derivatives can be accounted for in the future by inserting the appropriate terms in a straightforward manner.
- The outer boundary is spherical. It might be possible to generalize this condition but we have not tried this.
- The boundary conditions to the time derivative of the evolved variables are then given by:
The full boundary conditions, as applied to the time derivative of each evolved field are then given by:
\begin{align*} \partial_{t} \Psi&\to\partial_{t}\Psi + \lambda_\Psi n^i \mathcal{C}_i, \\ \partial_{t}\Pi&\to\partial_{t}\Pi-\left(\partial_t\Pi - \partial_t\Pi^{\mathrm{BC}}\right) +\gamma_2\lambda_\Psi n^i \mathcal{C}_i =\partial_t\Pi^{\mathrm{BC}} +\gamma_2\lambda_\Psi n^i \mathcal{C}_i, \\ \partial_{t}\Phi_i&\to\partial_{t}\Phi_i+ 2 \lambda_0 n^j \mathcal{C}_{ji}. \end{align*}
Member Function Documentation
◆ get_clone()
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overridevirtual |
Implements domain::BoundaryConditions::BoundaryCondition.
Member Data Documentation
◆ bc_type
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staticconstexpr |
◆ help
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staticconstexpr |
The documentation for this class was generated from the following file:
- src/Evolution/Systems/CurvedScalarWave/BoundaryConditions/ConstraintPreservingSphericalRadiation.hpp