Radial Sommerfeld boundary conditions for the m-mode metric perturbations. More...

#include <Sommerfeld.hpp>

Classes
struct	BlackHoleMass
struct	BlackHoleSpin
struct	HyperboloidalSlicing
struct	MModeNumber
struct	OrbitalRadius
struct	Order

Public Types
using	options
using	argument_tags = tmpl::list<Tags::Alpha, Tags::Beta, Tags::GammaRstar>
using	volume_tags = tmpl::list<>
using	argument_tags_linearized
using	volume_tags_linearized = tmpl::list<>

Public Member Functions
	Sommerfeld (const Sommerfeld &)=default
Sommerfeld &	operator= (const Sommerfeld &)=default
	Sommerfeld (Sommerfeld &&)=default
Sommerfeld &	operator= (Sommerfeld &&)=default
	Sommerfeld (double black_hole_mass, double black_hole_spin, double orbital_radius, int m_mode_number, bool hyperboloidal_slicing, int order)
double	black_hole_mass () const
double	black_hole_spin () const
double	orbital_radius () const
int	m_mode_number () const
bool	hyperboloidal_slicing () const
int	order () const
std::unique_ptr< domain::BoundaryConditions::BoundaryCondition >	get_clone () const override
std::vector< elliptic::BoundaryConditionType >	boundary_condition_types () const override
void	apply (gsl::not_null< tnsr::aa< ComplexDataVector, 3 > * > field, gsl::not_null< tnsr::aa< ComplexDataVector, 3 > * > n_dot_flux, const GradTensorType &deriv_field, const tnsr::I< ComplexDataVector, 2 > &alpha, const tnsr::aaBB< ComplexDataVector, 3 > &beta, const tnsr::aaBB< ComplexDataVector, 3 > &gamma_rstar) const
void	apply_linearized (gsl::not_null< tnsr::aa< ComplexDataVector, 3 > * > field_correction, gsl::not_null< tnsr::aa< ComplexDataVector, 3 > * > n_dot_flux_correction, const GradTensorType &deriv_field_correction, const tnsr::I< ComplexDataVector, 2 > &alpha, const tnsr::aaBB< ComplexDataVector, 3 > &beta, const tnsr::aaBB< ComplexDataVector, 3 > &gamma_rstar) const
void	pup (PUP::er &p) override
Public Member Functions inherited from elliptic::BoundaryConditions::BoundaryCondition< 2 >
	BoundaryCondition (const BoundaryCondition &)=default
	BoundaryCondition (BoundaryCondition &&)=default
BoundaryCondition &	operator= (const BoundaryCondition &)=default
BoundaryCondition &	operator= (BoundaryCondition &&)=default
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)

Static Public Attributes
static constexpr Options::String	help
Static Public Attributes inherited from elliptic::BoundaryConditions::BoundaryCondition< 2 >
static constexpr size_t	volume_dim

Friends
bool	operator== (const Sommerfeld &lhs, const Sommerfeld &rhs)

Detailed Description

Radial Sommerfeld boundary conditions for the m-mode metric perturbations.

These radial boundary conditions enforce an outgoing direction of radiation propagation. They apply both near the Kerr horizon (inner radial boundary) and at large distance (outer radial boundary):

\begin{equation}n_i F^i = i m \Omega \Psi_m \end{equation}

With hyperboloidal slicing (see HyperboloidalSlicing), the outgoing radiation condition above is replaced by a Neumann condition ( \(F^i = 0\)) at first order (Order = 1). At second order (Order = 2), it instead imposes

\begin{equation}\partial_{x^i}\left(\alpha^{i}\partial_{x^i}\Psi\right) = 0, \end{equation}

i.e. \(\gamma^{i}\cdot\partial_{x^i}\Psi = -\beta\cdot\Psi\), where \(x^i\) is the radial coordinate ( \(r_\star\) or \(r\), depending on PenetratingHorizon; see CircularOrbit). Since \(\gamma^{i}\) and \(\beta\) couple all tensor components of \(\Psi\) together, we invert this dense matrix in tensor-component space to solve for \(\partial_{x^i}\Psi\), then set the flux \(F^{i} = \alpha^{i}\partial_{x^i}\Psi\).

These boundary conditions currently assume a circular equatorial orbit.