ScalarWave Namespace Reference

Items related to evolving the scalar wave equation. More...

Namespaces
namespace	BoundaryConditions
	Boundary conditions for the scalar wave system.
namespace	BoundaryCorrections
	Boundary corrections/numerical fluxes.
namespace	Solutions
	Holds classes implementing a solution to the Euclidean wave equation \(0 = \frac{\partial^2 \Psi}{\partial t^2} - \nabla^2 \Psi\).
namespace	Tags
	Tags for the ScalarWave evolution system.

Classes
struct	ComputeNormalDotFluxes
	A relic of an old incorrect way of handling boundaries for non-conservative systems. More...
struct	System
struct	TimeDerivative
	Compute the time derivatives for scalar wave system. More...

Functions
template<size_t Dim>
std::array< DataVector, 4 >	characteristic_speeds (const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the scalar wave system. More...
template<size_t Dim>
void	characteristic_speeds (gsl::not_null< std::array< DataVector, 4 > * > char_speeds, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the scalar wave system. More...
template<size_t Dim>
Variables< tmpl::list< Tags::VPsi, Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > >	characteristic_fields (const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, Dim, Frame::Inertial > &phi, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Computes characteristic fields from evolved fields. More...
template<size_t Dim>
void	characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::VPsi, Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > > * > char_fields, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &psi, const Scalar< DataVector > &pi, const tnsr::i< DataVector, Dim, Frame::Inertial > &phi, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Computes characteristic fields from evolved fields. More...
template<size_t Dim>
Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< Dim > > >	evolved_fields_from_characteristic_fields (const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, Dim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute evolved fields from characteristic fields. More...
template<size_t Dim>
void	evolved_fields_from_characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< Dim > > > * > evolved_fields, const Scalar< DataVector > &gamma_2, const Scalar< DataVector > &v_psi, const tnsr::i< DataVector, Dim, Frame::Inertial > &v_zero, const Scalar< DataVector > &v_plus, const Scalar< DataVector > &v_minus, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute evolved fields from characteristic fields. More...
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)
	Compute 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)
	Compute 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)
	Compute 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)
	Compute the scalar-wave 2-index constraint. More...
template<size_t SpatialDim>
void	energy_density (gsl::not_null< Scalar< DataVector > * > result, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the energy density of the scalar wave system. More...
template<size_t SpatialDim>
Scalar< DataVector >	energy_density (const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the energy density of the scalar wave system. More...
template<size_t SpatialDim>
void	momentum_density (gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * > result, const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the momentum density of the scalar wave system. More...
template<size_t SpatialDim>
tnsr::i< DataVector, SpatialDim, Frame::Inertial >	momentum_density (const Scalar< DataVector > &pi, const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &phi)
	Computes the momentum density of the scalar wave system. More...

Detailed Description

Items related to evolving the scalar wave equation.

The equations of motion for the system augmented with constraint damping terms are given by Eq. (15), (23) and (24) of [102] (setting background spacetime to Minkowskian):

\begin{align*} \partial_t \psi =& -\Pi \\ \partial_t \Pi =& -\partial^i \Phi_i \\ \partial_t \Phi_i =& -\partial_i \Pi + \gamma_2 (\partial_i \psi - \Phi_i) \end{align*}

In our implementation here, to disable the constraint damping terms, set \(\gamma_2 = 0\).

Function Documentation

characteristic_fields() [1/2]

template<size_t Dim>

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

Computes characteristic fields from evolved fields.

Tags::CharacteristicFieldsCompute and Tags::EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system.

Tags::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 \({v}^{\hat \alpha}\) 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, \(\Phi_{i}=\partial_i \psi\) is an auxiliary variable, \(\Pi\) is a conjugate momentum, \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface along which the characteristic fields are defined.

Tags::EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(u_\alpha\) in terms of the characteristic fields. This uses the inverse of above relations:

\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 \(\lambda_{\hat \alpha}\) are computed by Tags::CharacteristicSpeedsCompute .

characteristic_fields() [2/2]

template<size_t Dim>

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

Computes characteristic fields from evolved fields.

Tags::CharacteristicFieldsCompute and Tags::EvolvedFieldsFromCharacteristicFieldsCompute convert between characteristic and evolved fields for the scalar-wave system.

Tags::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 \({v}^{\hat \alpha}\) 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, \(\Phi_{i}=\partial_i \psi\) is an auxiliary variable, \(\Pi\) is a conjugate momentum, \(\gamma_2\) is a constraint damping parameter, and \(n_k\) is the unit normal to the surface along which the characteristic fields are defined.

Tags::EvolvedFieldsFromCharacteristicFieldsCompute computes evolved fields \(u_\alpha\) in terms of the characteristic fields. This uses the inverse of above relations:

\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 \(\lambda_{\hat \alpha}\) are computed by Tags::CharacteristicSpeedsCompute .

characteristic_speeds() [1/2]

template<size_t Dim>

std::array< DataVector, 4 > ScalarWave::characteristic_speeds

(

const tnsr::i< DataVector, Dim, Frame::Inertial > &

unit_normal_one_form

)

Compute the characteristic speeds for the scalar wave system.

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 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 corresponding characteristic speeds \(\lambda_{\hat \alpha}\) are given in the text following Eq.(38) of [102] :

\begin{align*} \lambda_{\hat \psi} =& 0 \\ \lambda_{\hat 0} =& 0 \\ \lambda_{\hat \pm} =& \pm 1. \end{align*}

characteristic_speeds() [2/2]

template<size_t Dim>

void ScalarWave::characteristic_speeds	(	gsl::not_null< std::array< DataVector, 4 > * >	char_speeds,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	unit_normal_one_form
	)

Compute the characteristic speeds for the scalar wave system.

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 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 corresponding characteristic speeds \(\lambda_{\hat \alpha}\) are given in the text following Eq.(38) of [102] :

\begin{align*} \lambda_{\hat \psi} =& 0 \\ \lambda_{\hat 0} =& 0 \\ \lambda_{\hat \pm} =& \pm 1. \end{align*}

energy_density() [1/2]

template<size_t SpatialDim>

Scalar< DataVector > ScalarWave::energy_density	(	const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the energy density of the scalar wave system.

Below is the function used to calculate the energy density.

\begin{align*} \epsilon = \frac{1}{2}\left( \Pi^{2} + \abs{\Phi}^{2} \right) \end{align*}

energy_density() [2/2]

template<size_t SpatialDim>

void ScalarWave::energy_density	(	gsl::not_null< Scalar< DataVector > * >	result,
		const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the energy density of the scalar wave system.

Below is the function used to calculate the energy density.

\begin{align*} \epsilon = \frac{1}{2}\left( \Pi^{2} + \abs{\Phi}^{2} \right) \end{align*}

evolved_fields_from_characteristic_fields() [1/2]

template<size_t Dim>

Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< Dim > > > ScalarWave::evolved_fields_from_characteristic_fields	(	const Scalar< DataVector > &	gamma_2,
		const Scalar< DataVector > &	v_psi,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	v_zero,
		const Scalar< DataVector > &	v_plus,
		const Scalar< DataVector > &	v_minus,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	unit_normal_one_form
	)

Compute evolved fields from characteristic fields.

For expressions used here to compute evolved fields from characteristic ones, see Tags::CharacteristicFieldsCompute.

evolved_fields_from_characteristic_fields() [2/2]

template<size_t Dim>

void ScalarWave::evolved_fields_from_characteristic_fields	(	gsl::not_null< Variables< tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< Dim > > > * >	evolved_fields,
		const Scalar< DataVector > &	gamma_2,
		const Scalar< DataVector > &	v_psi,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	v_zero,
		const Scalar< DataVector > &	v_plus,
		const Scalar< DataVector > &	v_minus,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	unit_normal_one_form
	)

Compute evolved fields from characteristic fields.

For expressions used here to compute evolved fields from characteristic ones, see Tags::CharacteristicFieldsCompute.

momentum_density() [1/2]

template<size_t SpatialDim>

tnsr::i< DataVector, SpatialDim, Frame::Inertial > ScalarWave::momentum_density	(	const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the momentum density of the scalar wave system.

Below is the function used to calculate the momentum density.

\begin{align*} P_i = \Pi \times \Phi_i \end{align*}

momentum_density() [2/2]

template<size_t SpatialDim>

void ScalarWave::momentum_density	(	gsl::not_null< tnsr::i< DataVector, SpatialDim, Frame::Inertial > * >	result,
		const Scalar< DataVector > &	pi,
		const tnsr::i< DataVector, SpatialDim, Frame::Inertial > &	phi
	)

Computes the momentum density of the scalar wave system.

Below is the function used to calculate the momentum density.

\begin{align*} P_i = \Pi \times \Phi_i \end{align*}

one_index_constraint() [1/2]

template<size_t SpatialDim>

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

Compute 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 ScalarWave::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
	)

Compute 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 > ScalarWave::two_index_constraint

(

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

d_phi

)

Compute the scalar-wave 2-index constraint.

Details

Computes the scalar-wave 2-index constraint \(C_{ij} = \partial_i\Phi_j - \partial_j\Phi_i,\) where \(\Phi_{i} = \partial_i\psi\), that is given by Eq. (20) of [102]

Note

We do not support custom storage for antisymmetric tensors yet.

two_index_constraint() [2/2]

template<size_t SpatialDim>

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

Compute the scalar-wave 2-index constraint.

Details

Computes the scalar-wave 2-index constraint \(C_{ij} = \partial_i\Phi_j - \partial_j\Phi_i,\) where \(\Phi_{i} = \partial_i\psi\), that is given by Eq. (20) of [102]

Note

We do not support custom storage for antisymmetric tensors yet.