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| Robin (const Robin &)=default |
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Robin & | operator= (const Robin &)=default |
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| Robin (Robin &&)=default |
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Robin & | operator= (Robin &&)=default |
| std::unique_ptr< domain::BoundaryConditions::BoundaryCondition > | get_clone () const override |
| std::vector< elliptic::BoundaryConditionType > | boundary_condition_types () const override |
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void | apply (gsl::not_null< Scalar< DataVector > * > conformal_factor_minus_one, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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void | apply (gsl::not_null< Scalar< DataVector > * > conformal_factor_minus_one, gsl::not_null< Scalar< DataVector > * > lapse_times_conformal_factor_minus_one, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient, gsl::not_null< Scalar< DataVector > * > n_dot_lapse_times_conformal_factor_gradient, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::i< DataVector, 3 > &deriv_lapse_times_conformal_factor, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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void | apply (gsl::not_null< Scalar< DataVector > * > conformal_factor_minus_one, gsl::not_null< Scalar< DataVector > * > lapse_times_conformal_factor_minus_one, gsl::not_null< tnsr::I< DataVector, 3 > * > shift_excess, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient, gsl::not_null< Scalar< DataVector > * > n_dot_lapse_times_conformal_factor_gradient, gsl::not_null< tnsr::I< DataVector, 3 > * > n_dot_longitudinal_shift_excess, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::i< DataVector, 3 > &deriv_lapse_times_conformal_factor, const tnsr::iJ< DataVector, 3 > &deriv_shift_excess, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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void | apply_linearized (gsl::not_null< Scalar< DataVector > * > conformal_factor_correction, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient_correction, const tnsr::i< DataVector, 3 > &deriv_conformal_factor_correction, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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void | apply_linearized (gsl::not_null< Scalar< DataVector > * > conformal_factor_correction, gsl::not_null< Scalar< DataVector > * > lapse_times_conformal_factor_correction, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient_correction, gsl::not_null< Scalar< DataVector > * > n_dot_lapse_times_conformal_factor_gradient_correction, const tnsr::i< DataVector, 3 > &deriv_conformal_factor_correction, const tnsr::i< DataVector, 3 > &deriv_lapse_times_conformal_factor_correction, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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void | apply_linearized (gsl::not_null< Scalar< DataVector > * > conformal_factor_correction, gsl::not_null< Scalar< DataVector > * > lapse_times_conformal_factor_correction, gsl::not_null< tnsr::I< DataVector, 3 > * > shift_excess_correction, gsl::not_null< Scalar< DataVector > * > n_dot_conformal_factor_gradient_correction, gsl::not_null< Scalar< DataVector > * > n_dot_lapse_times_conformal_factor_gradient_correction, gsl::not_null< tnsr::I< DataVector, 3 > * > n_dot_longitudinal_shift_excess_correction, const tnsr::i< DataVector, 3 > &deriv_conformal_factor_correction, const tnsr::i< DataVector, 3 > &deriv_lapse_times_conformal_factor_correction, const tnsr::iJ< DataVector, 3 > &deriv_shift_excess_correction, const tnsr::I< DataVector, 3 > &x, const tnsr::i< DataVector, 3 > &face_normal) const |
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| BoundaryCondition (const BoundaryCondition &)=default |
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| BoundaryCondition (BoundaryCondition &&)=default |
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BoundaryCondition & | operator= (const BoundaryCondition &)=default |
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BoundaryCondition & | operator= (BoundaryCondition &&)=default |
| Public Member Functions inherited from domain::BoundaryConditions::BoundaryCondition |
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| BoundaryCondition (BoundaryCondition &&)=default |
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BoundaryCondition & | operator= (BoundaryCondition &&)=default |
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| BoundaryCondition (const BoundaryCondition &)=default |
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BoundaryCondition & | operator= (const BoundaryCondition &)=default |
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| BoundaryCondition (CkMigrateMessage *const msg) |
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| WRAPPED_PUPable_abstract (BoundaryCondition) |
template<
Xcts::Equations EnabledEquations>
class Xcts::BoundaryConditions::Robin< EnabledEquations >
Impose Robin boundary conditions at the outer boundary.
Impose \(\partial_r(r\psi)=0\), \(\partial_r(\alpha\psi)=0\), and \(\partial_r(r\beta_\mathrm{excess}^j)=0\) on the boundary. These Robin boundary conditions incur an error of order \(1/R^2\), where \(R\) is the outer radius of the domain. This allows to place the outer boundary much closer to the center than for Dirichlet boundary conditions (Xcts::BoundaryConditions::Flatness), which incur an error of order \(1/R\).
The Robin boundary conditions are imposed as Neumann-type as follows:
\begin{align*}(n^i \partial_i \psi)_N &= -\frac{\psi}{r}, \\
(n^i \partial_i (\alpha\psi))_N &= -\frac{\alpha\psi}{r}, \\
(n^i (L\beta)^{ij})_N = n^i (L\beta)^{ij} - \left[
\frac{\beta_\mathrm{excess}^j}{r}
+ \frac{1}{3} n^j \frac{n^i \beta_\mathrm{excess}^i}{r}
+ n^i \partial_i \beta_\mathrm{excess}^j
+ \frac{1}{3} n^j n^i n_k \partial_i \beta_\mathrm{excess}^k
\right]
\end{align*}
Here, the condition on the longitudinal shift is derived by imposing the Robin boundary condition \(n^i\partial_i\beta_\mathrm{excess}^j=-\beta_\mathrm{excess}^j/r\) only on the normal component of the shift gradient. To do this we can use the projection operator \(P_{ij}=\delta_{ij}-n_i n_j\) to set the shift gradient to \(\partial_i\beta_\mathrm{excess}^j=P_{ik}\partial_k\beta_\mathrm{excess}^j
-n_i \beta_\mathrm{excess}^j/r\) and then apply the longitudinal operator.
- Template Parameters
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| EnabledEquations | The subset of XCTS equations that are being solved |