Xcts::BoundaryConditions::Robin< EnabledEquations > Class Template Reference

Impose Robin boundary conditions at the outer boundary. More...

#include <Robin.hpp>

Public Types
using	options = implementation defined
using	argument_tags = implementation defined
using	volume_tags = implementation defined
using	argument_tags_linearized = implementation defined
using	volume_tags_linearized = implementation defined

Public Member Functions
	Robin (const Robin &)=default
Robin &	operator= (const Robin &)=default
	Robin (Robin &&)=default
Robin &	operator= (Robin &&)=default
std::unique_ptr< domain::BoundaryConditions::BoundaryCondition >	get_clone () const override
std::vector< elliptic::BoundaryConditionType >	boundary_condition_types () const override
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
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
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
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
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
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
Public Member Functions inherited from elliptic::BoundaryConditions::BoundaryCondition< 3 >
	BoundaryCondition (const BoundaryCondition &)=default
	BoundaryCondition (BoundaryCondition &&)=default
BoundaryCondition &	operator= (const BoundaryCondition &)=default
BoundaryCondition &	operator= (BoundaryCondition &&)=default
virtual std::vector< elliptic::BoundaryConditionType >	boundary_condition_types () const=0
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 Public Attributes inherited from elliptic::BoundaryConditions::BoundaryCondition< 3 >
static constexpr size_t	volume_dim

Detailed Description

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 ${\partial }_r(r{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j)=0$ 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:

$$\begin{array}{rl}(n^i{\partial }_i\psi {)}_N& =-\frac{\psi }{r},\\ (n^i{\partial }_i(\alpha \psi ){)}_N& =-\frac{\alpha \psi }{r},\\ (n^i(L\beta {)}^{ij}{)}_N=n^i(L\beta {)}^{ij}-[\frac{{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j}{r}+\frac{1}{3}{n}^j\frac{n^i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^i}{r}+n^i{\partial }_i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j+\frac{1}{3}{n}^j{n}^i{n}_k{\partial }_i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^k]\end{array}$$

Here, the condition on the longitudinal shift is derived by imposing the Robin boundary condition $n^i{\partial }_i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j=-{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j/r$ 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 ${\partial }_i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j=P_{ik}{\partial }_k{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j-n_i{\beta }_{\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^j/r$ and then apply the longitudinal operator.

Template Parameters

EnabledEquations

The subset of XCTS equations that are being solved

Member Function Documentation

boundary_condition_types()

template<Xcts::Equations EnabledEquations>

std::vector< elliptic::BoundaryConditionType > Xcts::BoundaryConditions::Robin< EnabledEquations >::boundary_condition_types ( ) const

overridevirtual

Implements elliptic::BoundaryConditions::BoundaryCondition< 3 >.

get_clone()

template<Xcts::Equations EnabledEquations>

std::unique_ptr< domain::BoundaryConditions::BoundaryCondition > Xcts::BoundaryConditions::Robin< EnabledEquations >::get_clone ( ) const

overridevirtual

Implements domain::BoundaryConditions::BoundaryCondition.

Member Data Documentation

help

template<Xcts::Equations EnabledEquations>

constexpr Options::String Xcts::BoundaryConditions::Robin< EnabledEquations >::help

staticconstexpr

Initial value:

=
      "Impose Robin boundary conditions at the outer boundary. "
      "They incur an error of order 1/R^2, where R is the outer radius."

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

• src/Elliptic/Systems/Xcts/BoundaryConditions/Robin.hpp