gh::BoundaryConditions::Bjorhus Namespace Reference

Detailed implementation of Bjorhus-type boundary corrections. More...

Functions
template<size_t VolumeDim, typename DataType >
void	constraint_preserving_bjorhus_corrections_dt_v_psi (gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * > bc_dt_v_psi, const tnsr::I< DataType, VolumeDim, Frame::Inertial > &unit_interface_normal_vector, const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &three_index_constraint, const std::array< DataType, 4 > &char_speeds)
	Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{\psi }$. More...
template<size_t VolumeDim, typename DataType >
void	constraint_preserving_bjorhus_corrections_dt_v_zero (gsl::not_null< tnsr::iaa< DataType, VolumeDim, Frame::Inertial > * > bc_dt_v_zero, const tnsr::I< DataType, VolumeDim, Frame::Inertial > &unit_interface_normal_vector, const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &four_index_constraint, const std::array< DataType, 4 > &char_speeds)
	Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{iab}^0$. More...
template<size_t VolumeDim, typename DataType >
void	constraint_preserving_bjorhus_corrections_dt_v_minus (gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * > bc_dt_v_minus, const Scalar< DataType > &gamma2, const tnsr::I< DataType, VolumeDim, Frame::Inertial > &inertial_coords, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &incoming_null_one_form, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &outgoing_null_one_form, const tnsr::A< DataType, VolumeDim, Frame::Inertial > &incoming_null_vector, const tnsr::A< DataType, VolumeDim, Frame::Inertial > &outgoing_null_vector, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &projection_ab, const tnsr::Ab< DataType, VolumeDim, Frame::Inertial > &projection_Ab, const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &projection_AB, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &char_projected_rhs_dt_v_psi, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &char_projected_rhs_dt_v_minus, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &constraint_char_zero_plus, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &constraint_char_zero_minus, const std::array< DataType, 4 > &char_speeds)
	Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{-}$. More...
template<size_t VolumeDim, typename DataType >
void	constraint_preserving_physical_bjorhus_corrections_dt_v_minus (gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * > bc_dt_v_minus, const Scalar< DataType > &gamma2, const tnsr::I< DataType, VolumeDim, Frame::Inertial > &inertial_coords, const tnsr::i< DataType, VolumeDim, Frame::Inertial > &unit_interface_normal_one_form, const tnsr::I< DataType, VolumeDim, Frame::Inertial > &unit_interface_normal_vector, const tnsr::A< DataType, VolumeDim, Frame::Inertial > &spacetime_unit_normal_vector, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &incoming_null_one_form, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &outgoing_null_one_form, const tnsr::A< DataType, VolumeDim, Frame::Inertial > &incoming_null_vector, const tnsr::A< DataType, VolumeDim, Frame::Inertial > &outgoing_null_vector, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &projection_ab, const tnsr::Ab< DataType, VolumeDim, Frame::Inertial > &projection_Ab, const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &projection_AB, const tnsr::II< DataType, VolumeDim, Frame::Inertial > &inverse_spatial_metric, const tnsr::ii< DataType, VolumeDim, Frame::Inertial > &extrinsic_curvature, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &spacetime_metric, const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &inverse_spacetime_metric, const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &three_index_constraint, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &char_projected_rhs_dt_v_psi, const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &char_projected_rhs_dt_v_minus, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &constraint_char_zero_plus, const tnsr::a< DataType, VolumeDim, Frame::Inertial > &constraint_char_zero_minus, const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &phi, const tnsr::ijaa< DataType, VolumeDim, Frame::Inertial > &d_phi, const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &d_pi, const std::array< DataType, 4 > &char_speeds)
	Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{-}$. More...

Detailed Description

Detailed implementation of Bjorhus-type boundary corrections.

Function Documentation

constraint_preserving_bjorhus_corrections_dt_v_minus()

template<size_t VolumeDim, typename DataType >

void gh::BoundaryConditions::Bjorhus::constraint_preserving_bjorhus_corrections_dt_v_minus	(	gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * >	bc_dt_v_minus,
		const Scalar< DataType > &	gamma2,
		const tnsr::I< DataType, VolumeDim, Frame::Inertial > &	inertial_coords,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	incoming_null_one_form,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	outgoing_null_one_form,
		const tnsr::A< DataType, VolumeDim, Frame::Inertial > &	incoming_null_vector,
		const tnsr::A< DataType, VolumeDim, Frame::Inertial > &	outgoing_null_vector,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	projection_ab,
		const tnsr::Ab< DataType, VolumeDim, Frame::Inertial > &	projection_Ab,
		const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &	projection_AB,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	char_projected_rhs_dt_v_psi,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	char_projected_rhs_dt_v_minus,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	constraint_char_zero_plus,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	constraint_char_zero_minus,
		const std::array< DataType, 4 > &	char_speeds
	)

Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{-}$.

Details

In the Bjorhus scheme, the time derivatives of evolved variables are characteristic projected. Constraint-preserving correction terms $T_{ab}^{\mathrm{C}}$, Sommerfeld condition terms for gauge degrees of freedom $T_{ab}^{\mathrm{G}}$, and terms constraining physical degrees of freedom $T_{ab}^{\mathrm{P}}$ are added here to the resulting characteristic (time-derivative) field:

$$\begin{array}{}\text{(1)}& \mathrm{\Delta }{\partial }_t{v}_{ab}^{-}=T_{ab}^{\mathrm{C}}+T_{ab}^{\mathrm{G}}+T_{ab}^{\mathrm{P}}.\end{array}$$

These terms are given by Eq. (64) of [127] :

$$\begin{array}{rcl}\text{(2)}& T_{ab}^{\mathrm{C}}& =& \frac{1}{2}(2k^c{k}^d{l}_a{l}_b-k^c{l}_b{P}_a^d-k^c{l}_a{P}_b^d-k^d{l}_b{P}_a^c-k^d{l}_a{P}_b^c+P^{cd}{P}_{ab}){\partial }_t{v}_{cd}^{-}& & +\frac{1}{\sqrt{2}}{\lambda }_{-}{c}_c^{\hat{0}-}(l_a{l}_b{k}^c+P_{ab}{l}^c-P_b^c{l}_a-P_a^c{l}_b)\end{array}$$

where $l^a(l_a)$ is the outgoing null vector (one-form), $k^a(k_a)$ is the incoming null vector (one-form), $P_{ab},P^{ab},P_b^a$ are spacetime projection operators defined in transverse_projection_operator(), ${\partial }_t{v}_{ab}^{-}$ is the characteristic projected time derivative of evolved variables (corresponding to the $v^{-}$ field), and $c_a^{\hat{0}\pm }$ are characteristic modes of the constraint evolution system:

$$\begin{array}{r}{c}_a^{\hat{0}\pm }=F_a\mp n^k{C}_{ka},\end{array}$$

where $F_a$ is the generalized-harmonic (GH) F constraint [Eq. (43) of [127]], $C_{ka}$ is the GH 2-index constraint [Eq. (44) of [127]], and $n^k$ is the unit spatial normal to the outer boundary. Boundary correction terms that prevent strong reflections of gauge perturbations are given by Eq. (25) of [173] :

$$\begin{array}{}\text{(3)}& T_{ab}^{\mathrm{G}}=(k_a{P}_b^c{l}^d+k_b{P}_a^c{l}^d-(k_a{l}_b{k}^c{l}^d+k_b{l}_a{k}_c{l}^d+k_a{k}_b{l}^c{l}^d))({\gamma }_2-\frac{1}{r}){\partial }_t{v}_{cd}^{\psi }\end{array}$$

where $r$ is the radial coordinate at the outer boundary, which is assumed to be spherical, ${\gamma }_2$ is a GH constraint damping parameter, and ${\partial }_t{v}_{ab}^{\psi }$ is the characteristic projected time derivative of evolved variables (corresponding to the $v^{\psi }$ field). Finally, we constrain physical degrees of freedom using corrections from Eq. (68) of [127] :

$$\begin{array}{}\text{(4)}& T_{ab}^{\mathrm{P}}=(P_a^c{P}_b^d-\frac{1}{2}{P}_{ab}{P}^{cd})({\partial }_t{v}_{cd}^{-}+{\lambda }_{-}(U_{cd}^{3-}-{\gamma }_2{n}^i{C}_{icd})),\end{array}$$

where $C_{icd}$ is the GH 3-index constraint [c.f. Eq. (26) of [127], see also three_index_constraint()], and

$$\begin{array}{r}{U}_{ab}^{3-}=2P_a^i{P}_b^j{U}_{ij}^{8-}\end{array}$$

is the inward propagating characteristic mode of the Weyl tensor evolution $U_{ab}^{8-}$ [c.f. Eq. 75 of [110] ] projected onto the outer boundary using the spatial-spacetime projection operators $P_a^i$. Note that (A) the covariant derivative of extrinsic curvature needed to get the Weyl propagating modes is calculated substituting the evolved variable ${\mathrm{\Phi }}_{iab}$ for spatial derivatives of the spacetime metric; and (B) the spatial Ricci tensor used in the same calculation is also calculated using the same substituion, and also includes corrections proportional to the GH 4-index constraint $C_{ijab}$ [c.f. Eq. (45) of [127] , see also four_index_constraint()]:

$$\begin{array}{r}{R}_{ij}\to R_{ij}+C_{(iklj)}+\frac{1}{2}{n}^k{q}^a{C}_{(ikj)a},\end{array}$$

where $q^a$ is the future-directed spacetime normal vector.

constraint_preserving_bjorhus_corrections_dt_v_psi()

template<size_t VolumeDim, typename DataType >

void gh::BoundaryConditions::Bjorhus::constraint_preserving_bjorhus_corrections_dt_v_psi	(	gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * >	bc_dt_v_psi,
		const tnsr::I< DataType, VolumeDim, Frame::Inertial > &	unit_interface_normal_vector,
		const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &	three_index_constraint,
		const std::array< DataType, 4 > &	char_speeds
	)

Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{\psi }$.

Details

In the Bjorhus scheme, the time derivatives of evolved variables are characteristic projected. A constraint-preserving correction term is added here to the resulting characteristic (time-derivative) field:

$$\begin{array}{}\text{(5)}& \mathrm{\Delta }{\partial }_t{v}_{ab}^{\psi }={\lambda }_{\psi }{n}^i{C}_{iab}\end{array}$$

where $n^i$ is the local unit normal to the external boundary, $C_{iab}={\partial }_i{\psi }_{ab}-{\mathrm{\Phi }}_{iab}$ is the three-index constraint, and ${\lambda }_{\psi }$ is the characteristic speed of the field $v_{ab}^{\psi }$.

constraint_preserving_bjorhus_corrections_dt_v_zero()

template<size_t VolumeDim, typename DataType >

void gh::BoundaryConditions::Bjorhus::constraint_preserving_bjorhus_corrections_dt_v_zero	(	gsl::not_null< tnsr::iaa< DataType, VolumeDim, Frame::Inertial > * >	bc_dt_v_zero,
		const tnsr::I< DataType, VolumeDim, Frame::Inertial > &	unit_interface_normal_vector,
		const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &	four_index_constraint,
		const std::array< DataType, 4 > &	char_speeds
	)

Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{iab}^0$.

Details

In the Bjorhus scheme, the time derivatives of evolved variables are characteristic projected. A constraint-preserving correction term is added here to the resulting characteristic (time-derivative) field $v_{iab}^0$:

$$\begin{array}{}\text{(6)}& \mathrm{\Delta }{\partial }_t{v}_{iab}^0={\lambda }_0{n}^j{C}_{jiab}\end{array}$$

where $n^i$ is the local unit normal to the external boundary, $C_{ijab}={\partial }_i{\mathrm{\Phi }}_{jab}-{\partial }_j{\mathrm{\Phi }}_{iab}$ is the four-index constraint, and ${\lambda }_0$ is the characteristic speed of the field $v_{iab}^0$.

Note

In 3D, only the non-zero and unique components of the four-index constraint are stored, as ${\hat{C}}_{iab}={ϵ}_i^{jk}{C}_{jkab}$. In 2D the input expected here is ${\hat{C}}_{0ab}=C_{01ab},{\hat{C}}_{1ab}=C_{10ab}$, and in 1D ${\hat{C}}_{0ab}=0$.

constraint_preserving_physical_bjorhus_corrections_dt_v_minus()

template<size_t VolumeDim, typename DataType >

void gh::BoundaryConditions::Bjorhus::constraint_preserving_physical_bjorhus_corrections_dt_v_minus	(	gsl::not_null< tnsr::aa< DataType, VolumeDim, Frame::Inertial > * >	bc_dt_v_minus,
		const Scalar< DataType > &	gamma2,
		const tnsr::I< DataType, VolumeDim, Frame::Inertial > &	inertial_coords,
		const tnsr::i< DataType, VolumeDim, Frame::Inertial > &	unit_interface_normal_one_form,
		const tnsr::I< DataType, VolumeDim, Frame::Inertial > &	unit_interface_normal_vector,
		const tnsr::A< DataType, VolumeDim, Frame::Inertial > &	spacetime_unit_normal_vector,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	incoming_null_one_form,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	outgoing_null_one_form,
		const tnsr::A< DataType, VolumeDim, Frame::Inertial > &	incoming_null_vector,
		const tnsr::A< DataType, VolumeDim, Frame::Inertial > &	outgoing_null_vector,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	projection_ab,
		const tnsr::Ab< DataType, VolumeDim, Frame::Inertial > &	projection_Ab,
		const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &	projection_AB,
		const tnsr::II< DataType, VolumeDim, Frame::Inertial > &	inverse_spatial_metric,
		const tnsr::ii< DataType, VolumeDim, Frame::Inertial > &	extrinsic_curvature,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	spacetime_metric,
		const tnsr::AA< DataType, VolumeDim, Frame::Inertial > &	inverse_spacetime_metric,
		const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &	three_index_constraint,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	char_projected_rhs_dt_v_psi,
		const tnsr::aa< DataType, VolumeDim, Frame::Inertial > &	char_projected_rhs_dt_v_minus,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	constraint_char_zero_plus,
		const tnsr::a< DataType, VolumeDim, Frame::Inertial > &	constraint_char_zero_minus,
		const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &	phi,
		const tnsr::ijaa< DataType, VolumeDim, Frame::Inertial > &	d_phi,
		const tnsr::iaa< DataType, VolumeDim, Frame::Inertial > &	d_pi,
		const std::array< DataType, 4 > &	char_speeds
	)

Computes the expression needed to set boundary conditions on the time derivative of the characteristic field $v_{ab}^{-}$.

Details

In the Bjorhus scheme, the time derivatives of evolved variables are characteristic projected. Constraint-preserving correction terms $T_{ab}^{\mathrm{C}}$, Sommerfeld condition terms for gauge degrees of freedom $T_{ab}^{\mathrm{G}}$, and terms constraining physical degrees of freedom $T_{ab}^{\mathrm{P}}$ are added here to the resulting characteristic (time-derivative) field:

$$\begin{array}{}\text{(7)}& \mathrm{\Delta }{\partial }_t{v}_{ab}^{-}=T_{ab}^{\mathrm{C}}+T_{ab}^{\mathrm{G}}+T_{ab}^{\mathrm{P}}.\end{array}$$

These terms are given by Eq. (64) of [127] :

$$\begin{array}{rcl}\text{(8)}& T_{ab}^{\mathrm{C}}& =& \frac{1}{2}(2k^c{k}^d{l}_a{l}_b-k^c{l}_b{P}_a^d-k^c{l}_a{P}_b^d-k^d{l}_b{P}_a^c-k^d{l}_a{P}_b^c+P^{cd}{P}_{ab}){\partial }_t{v}_{cd}^{-}& & +\frac{1}{\sqrt{2}}{\lambda }_{-}{c}_c^{\hat{0}-}(l_a{l}_b{k}^c+P_{ab}{l}^c-P_b^c{l}_a-P_a^c{l}_b)\end{array}$$

where $l^a(l_a)$ is the outgoing null vector (one-form), $k^a(k_a)$ is the incoming null vector (one-form), $P_{ab},P^{ab},P_b^a$ are spacetime projection operators defined in transverse_projection_operator(), ${\partial }_t{v}_{ab}^{-}$ is the characteristic projected time derivative of evolved variables (corresponding to the $v^{-}$ field), and $c_a^{\hat{0}\pm }$ are characteristic modes of the constraint evolution system:

$$\begin{array}{r}{c}_a^{\hat{0}\pm }=F_a\mp n^k{C}_{ka},\end{array}$$

where $F_a$ is the generalized-harmonic (GH) F constraint [Eq. (43) of [127]], $C_{ka}$ is the GH 2-index constraint [Eq. (44) of [127]], and $n^k$ is the unit spatial normal to the outer boundary. Boundary correction terms that prevent strong reflections of gauge perturbations are given by Eq. (25) of [173] :

$$\begin{array}{}\text{(9)}& T_{ab}^{\mathrm{G}}=(k_a{P}_b^c{l}^d+k_b{P}_a^c{l}^d-(k_a{l}_b{k}^c{l}^d+k_b{l}_a{k}_c{l}^d+k_a{k}_b{l}^c{l}^d))({\gamma }_2-\frac{1}{r}){\partial }_t{v}_{cd}^{\psi }\end{array}$$

where $r$ is the radial coordinate at the outer boundary, which is assumed to be spherical, ${\gamma }_2$ is a GH constraint damping parameter, and ${\partial }_t{v}_{ab}^{\psi }$ is the characteristic projected time derivative of evolved variables (corresponding to the $v^{\psi }$ field). Finally, we constrain physical degrees of freedom using corrections from Eq. (68) of [127] :

$$\begin{array}{}\text{(10)}& T_{ab}^{\mathrm{P}}=(P_a^c{P}_b^d-\frac{1}{2}{P}_{ab}{P}^{cd})({\partial }_t{v}_{cd}^{-}+{\lambda }_{-}(U_{cd}^{3-}-{\gamma }_2{n}^i{C}_{icd})),\end{array}$$

where $C_{icd}$ is the GH 3-index constraint [c.f. Eq. (26) of [127], see also three_index_constraint()], and

$$\begin{array}{r}{U}_{ab}^{3-}=2P_a^i{P}_b^j{U}_{ij}^{8-}\end{array}$$

is the inward propagating characteristic mode of the Weyl tensor evolution $U_{ab}^{8-}$ [c.f. Eq. 75 of [110] ] projected onto the outer boundary using the spatial-spacetime projection operators $P_a^i$. Note that (A) the covariant derivative of extrinsic curvature needed to get the Weyl propagating modes is calculated substituting the evolved variable ${\mathrm{\Phi }}_{iab}$ for spatial derivatives of the spacetime metric; and (B) the spatial Ricci tensor used in the same calculation is also calculated using the same substituion, and also includes corrections proportional to the GH 4-index constraint $C_{ijab}$ [c.f. Eq. (45) of [127] , see also four_index_constraint()]:

$$\begin{array}{r}{R}_{ij}\to R_{ij}+C_{(iklj)}+\frac{1}{2}{n}^k{q}^a{C}_{(ikj)a},\end{array}$$

where $q^a$ is the future-directed spacetime normal vector.