CurvedScalarWave Namespace Reference

Items related to evolving a scalar wave on a curved background. More...

Namespaces
namespace	Actions
	Actions for the curved scalar wave system.
namespace	AnalyticData
	Holds classes implementing analytic data for the CurvedScalarWave system.
namespace	BoundaryConditions
	Boundary conditions for the curved scalar wave system.
namespace	BoundaryCorrections
	Boundary corrections/numerical fluxes.
namespace	Initialization
	Utilities for initializing the curved scalar wave system.
namespace	OptionTags
	Option tags for the curved scalar wave system.
namespace	Tags
	Tags for the curved scalar wave system.
namespace	Worldtube
	The set of utilities for performing CurvedScalarWave evolution with a worldtube excision scheme.

Classes
struct	CharacteristicFieldsCompute
struct	CharacteristicSpeedsCompute
struct	EvolvedFieldsFromCharacteristicFieldsCompute
class	NumericInitialData
	Numeric initial data loaded from volume data files. More...
struct	System
struct	TimeDerivative
	Compute the time derivative of the evolved variables of the first-order scalar wave system on a curved background. More...

Functions
template<size_t SpatialDim>
std::array< DataVector, 4 >	characteristic_speeds (const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the scalar wave system in curved spacetime. More...
template<size_t SpatialDim>
void	characteristic_speeds (gsl::not_null< std::array< DataVector, 4 > * > char_speeds, const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the scalar wave system in curved spacetime. More...
template<size_t SpatialDim>
void	characteristic_speeds (gsl::not_null< tnsr::a< DataVector, 3, Frame::Inertial > * > char_speeds, const Scalar< DataVector > &gamma_1, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &shift, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the scalar wave system in curved spacetime. More...
template<size_t SpatialDim>
Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > >	characteristic_fields (const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &unit_normal_vector)
	Computes characteristic fields from evolved fields. More...
template<size_t SpatialDim>
void	characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > > * > char_fields, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &unit_normal_vector)
	Computes characteristic fields from evolved fields. More...
template<size_t SpatialDim>
void	characteristic_fields (const gsl::not_null< Scalar< DataVector > * > &v_psi, const gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > &v_zero, const gsl::not_null< Scalar< DataVector > * > &v_plus, const gsl::not_null< Scalar< DataVector > * > &v_minus, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form, const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &unit_normal_vector)
	Computes characteristic fields from evolved fields. More...
template<size_t SpatialDim>
Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< SpatialDim > > >	evolved_fields_from_characteristic_fields (const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	For expressions used here to compute evolved fields from characteristic ones, see CharacteristicFieldsCompute.
template<size_t SpatialDim>
void	evolved_fields_from_characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< SpatialDim > > > * > evolved_fields, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	For expressions used here to compute evolved fields from characteristic ones, see CharacteristicFieldsCompute.
template<size_t SpatialDim>
void	evolved_fields_from_characteristic_fields (gsl::not_null< Scalar< DataVector > * > psi, gsl::not_null< Scalar< DataVector > * > pi, gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > phi, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &unit_normal_one_form)
	For expressions used here to compute evolved fields from characteristic ones, see CharacteristicFieldsCompute.
template<size_t SpatialDim>
tnsr::i< DataVector, SpatialDim, Frame::Inertial >	one_index_constraint (const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &d_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the scalar-wave one-index constraint. More...
template<size_t SpatialDim>
void	one_index_constraint (gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > constraint, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &d_psi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the scalar-wave one-index constraint. More...
template<size_t SpatialDim>
tnsr::ij< DataVector, SpatialDim, Frame::Inertial >	two_index_constraint (const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &d_phi)
	Computes the scalar-wave 2-index constraint. More...
template<size_t SpatialDim>
void	two_index_constraint (gsl::not_null< tnsr::ij< DataVector, SpatialDim, Frame::Inertial > * > constraint, const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &d_phi)
	Computes the scalar-wave 2-index constraint. More...

Detailed Description

Items related to evolving a scalar wave on a curved background.

Function Documentation

characteristic_fields() [1/3]

template<size_t SpatialDim>

void CurvedScalarWave::characteristic_fields	(	const gsl::not_null< Scalar< DataVector > * > &	v_psi,
		const gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > &	v_zero,
		const gsl::not_null< Scalar< DataVector > * > &	v_plus,
		const gsl::not_null< Scalar< DataVector > * > &	v_minus,
		const Scalar< DataVector > &	gamma_2,
		const Scalar< DataVector > &	psi,
		const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_vector
	)

Computes characteristic fields from evolved fields.

CharacteristicFieldsCompute and EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system in curved spacetime.

CharacteristicFieldsCompute computes characteristic fields as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] . Their names used here differ from this paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The characteristic fields \(u\) are given in terms of the evolved fields by Eq. (33) - (35) of [102], respectively:

\begin{align*} v^{\hat \psi} =& \psi \\ v^{\hat 0}_{i} =& (\delta^k_i - n_i n^k) \Phi_{k} := P^k_i \Phi_{k} \\ v^{\hat \pm} =& \Pi \pm n^i \Phi_{i} - \gamma_2\psi \end{align*}

where \(\psi\) is the scalar field, \(\Pi\) and \(\Phi_{i}\) are evolved fields introduced by first derivatives of \(\psi\), \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface.

EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(w\) in terms of the characteristic fields. This uses the inverse of above relations (c.f. Eq. (36) - (38) of [102] ):

\begin{align*} \psi =& v^{\hat \psi}, \\ \Pi =& \frac{1}{2}(v^{\hat +} + v^{\hat -}) + \gamma_2 v^{\hat \psi}, \\ \Phi_{i} =& \frac{1}{2}(v^{\hat +} - v^{\hat -}) n_i + v^{\hat 0}_{i}. \end{align*}

The corresponding characteristic speeds \(\lambda\) are computed by CharacteristicSpeedsCompute .

characteristic_fields() [2/3]

template<size_t SpatialDim>

Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > > CurvedScalarWave::characteristic_fields	(	const Scalar< DataVector > &	gamma_2,
		const Scalar< DataVector > &	psi,
		const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_vector
	)

Computes characteristic fields from evolved fields.

CharacteristicFieldsCompute and EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system in curved spacetime.

CharacteristicFieldsCompute computes characteristic fields as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] . Their names used here differ from this paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The characteristic fields \(u\) are given in terms of the evolved fields by Eq. (33) - (35) of [102], respectively:

\begin{align*} v^{\hat \psi} =& \psi \\ v^{\hat 0}_{i} =& (\delta^k_i - n_i n^k) \Phi_{k} := P^k_i \Phi_{k} \\ v^{\hat \pm} =& \Pi \pm n^i \Phi_{i} - \gamma_2\psi \end{align*}

where \(\psi\) is the scalar field, \(\Pi\) and \(\Phi_{i}\) are evolved fields introduced by first derivatives of \(\psi\), \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface.

EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(w\) in terms of the characteristic fields. This uses the inverse of above relations (c.f. Eq. (36) - (38) of [102] ):

\begin{align*} \psi =& v^{\hat \psi}, \\ \Pi =& \frac{1}{2}(v^{\hat +} + v^{\hat -}) + \gamma_2 v^{\hat \psi}, \\ \Phi_{i} =& \frac{1}{2}(v^{\hat +} - v^{\hat -}) n_i + v^{\hat 0}_{i}. \end{align*}

The corresponding characteristic speeds \(\lambda\) are computed by CharacteristicSpeedsCompute .

characteristic_fields() [3/3]

template<size_t SpatialDim>

void CurvedScalarWave::characteristic_fields	(	gsl::not_null< Variables< tmpl::list< Tags::VPsi, Tags::VZero< SpatialDim >, Tags::VPlus, Tags::VMinus > > * >	char_fields,
		const Scalar< DataVector > &	gamma_2,
		const Scalar< DataVector > &	psi,
		const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_vector
	)

Computes characteristic fields from evolved fields.

CharacteristicFieldsCompute and EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system in curved spacetime.

CharacteristicFieldsCompute computes characteristic fields as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] . Their names used here differ from this paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The characteristic fields \(u\) are given in terms of the evolved fields by Eq. (33) - (35) of [102], respectively:

\begin{align*} v^{\hat \psi} =& \psi \\ v^{\hat 0}_{i} =& (\delta^k_i - n_i n^k) \Phi_{k} := P^k_i \Phi_{k} \\ v^{\hat \pm} =& \Pi \pm n^i \Phi_{i} - \gamma_2\psi \end{align*}

where \(\psi\) is the scalar field, \(\Pi\) and \(\Phi_{i}\) are evolved fields introduced by first derivatives of \(\psi\), \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface.

EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(w\) in terms of the characteristic fields. This uses the inverse of above relations (c.f. Eq. (36) - (38) of [102] ):

\begin{align*} \psi =& v^{\hat \psi}, \\ \Pi =& \frac{1}{2}(v^{\hat +} + v^{\hat -}) + \gamma_2 v^{\hat \psi}, \\ \Phi_{i} =& \frac{1}{2}(v^{\hat +} - v^{\hat -}) n_i + v^{\hat 0}_{i}. \end{align*}

The corresponding characteristic speeds \(\lambda\) are computed by CharacteristicSpeedsCompute .

characteristic_speeds() [1/3]

template<size_t SpatialDim>

std::array< DataVector, 4 > CurvedScalarWave::characteristic_speeds	(	const Scalar< DataVector > &	gamma_1,
		const Scalar< DataVector > &	lapse,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	shift,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form
	)

Compute the characteristic speeds for the scalar wave system in curved spacetime.

Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [102] :

\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}

where \(n_k\) is the unit normal to the surface.

characteristic_speeds() [2/3]

template<size_t SpatialDim>

void CurvedScalarWave::characteristic_speeds	(	gsl::not_null< std::array< DataVector, 4 > * >	char_speeds,
		const Scalar< DataVector > &	gamma_1,
		const Scalar< DataVector > &	lapse,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	shift,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form
	)

Compute the characteristic speeds for the scalar wave system in curved spacetime.

Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [102] :

\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}

where \(n_k\) is the unit normal to the surface.

characteristic_speeds() [3/3]

template<size_t SpatialDim>

void CurvedScalarWave::characteristic_speeds	(	gsl::not_null< tnsr::a< DataVector, 3, Frame::Inertial > * >	char_speeds,
		const Scalar< DataVector > &	gamma_1,
		const Scalar< DataVector > &	lapse,
		const tnsr::I< DataVector, SpatialDim, Frame::Inertial > &	shift,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	unit_normal_one_form
	)

Compute the characteristic speeds for the scalar wave system in curved spacetime.

Computes the speeds as described in "Optimal constraint projection for hyperbolic evolution systems" by Holst et. al [102] [see text following Eq. (32)]. The characteristic fields' names used here are similar to the paper:

\begin{align*} \mathrm{SpECTRE} && \mathrm{Holst} \\ v^{\hat \psi} && Z^1 \\ v^{\hat 0}_{i} && Z^{2}_{i} \\ v^{\hat \pm} && u^{1\pm} \end{align*}

The corresponding characteristic speeds \(\lambda\) are given in the text following Eq. (38) of [102] :

\begin{align*} \lambda_{\hat \psi} =& -(1 + \gamma_1) n_k N^k \\ \lambda_{\hat 0} =& -n_k N^k \\ \lambda_{\hat \pm} =& -n_k N^k \pm N \end{align*}

where \(n_k\) is the unit normal to the surface.

one_index_constraint() [1/2]

template<size_t SpatialDim>

tnsr::i< DataVector, SpatialDim, Frame::Inertial > CurvedScalarWave::one_index_constraint	(	const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	d_psi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the scalar-wave one-index constraint.

Details

Computes the scalar-wave one-index constraint, \(C_{i} = \partial_i\psi - \Phi_{i},\) which is given by Eq. (19) of [102]

one_index_constraint() [2/2]

template<size_t SpatialDim>

void CurvedScalarWave::one_index_constraint	(	gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * >	constraint,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	d_psi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the scalar-wave one-index constraint.

Details

Computes the scalar-wave one-index constraint, \(C_{i} = \partial_i\psi - \Phi_{i},\) which is given by Eq. (19) of [102]

two_index_constraint() [1/2]

template<size_t SpatialDim>

tnsr::ij< DataVector, SpatialDim, Frame::Inertial > CurvedScalarWave::two_index_constraint

(

const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &

d_phi

)

Computes the scalar-wave 2-index constraint.

Details

Computes the scalar-wave 2-index FOSH constraint [Eq. (20) of [102]],

\begin{eqnarray} C_{ij} &\equiv& \partial_i \Phi_j - \partial_j \Phi_i \end{eqnarray}

where \(\Phi_{i} = \partial_i\psi\); and \(\psi\) is the scalar field.

two_index_constraint() [2/2]

template<size_t SpatialDim>

void CurvedScalarWave::two_index_constraint	(	gsl::not_null< tnsr::ij< DataVector, SpatialDim, Frame::Inertial > * >	constraint,
		const tnsr::ij< DataVector, SpatialDim, Frame::Inertial > &	d_phi
	)

Computes the scalar-wave 2-index constraint.

Details

Computes the scalar-wave 2-index FOSH constraint [Eq. (20) of [102]],

\begin{eqnarray} C_{ij} &\equiv& \partial_i \Phi_j - \partial_j \Phi_i \end{eqnarray}

where \(\Phi_{i} = \partial_i\psi\); and \(\psi\) is the scalar field.