SecondOrderScalarWave Namespace Reference

Items related to evolving the second-order scalar wave equations. More...

Namespaces
namespace	Solutions
	Holds analytic solutions to the second-order-in-space formulation of the Euclidean wave equation.
namespace	Tags
	Tags for the second-order ScalarWave evolution system.

Classes
struct	System
struct	TimeDerivative
	Compute the time derivatives for the second-order scalar wave system. More...

Functions
template<size_t Dim>
std::array< DataVector, 3 >	characteristic_speeds (const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the second-order scalar wave system.
template<size_t Dim>
void	characteristic_speeds (gsl::not_null< std::array< DataVector, 3 > * > char_speeds, const tnsr::i< DataVector, Dim, Frame::Inertial > &unit_normal_one_form)
	Compute the characteristic speeds for the second-order scalar wave system.
template<size_t Dim>
Variables< tmpl::list< Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > >	characteristic_fields (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 for the second-order scalar wave system.
template<size_t Dim>
void	characteristic_fields (gsl::not_null< Variables< tmpl::list< Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > > * > char_fields, 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 for the second-order scalar wave system.
template<size_t Dim>
Variables< tmpl::list< Tags::Pi, Tags::Phi< Dim > > >	fields_from_inverse_characteristic_transform (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)
	Reconstruct the fields \((\Pi, \Phi_i)\) from the characteristic fields.
template<size_t Dim>
void	fields_from_inverse_characteristic_transform (gsl::not_null< Variables< tmpl::list< Tags::Pi, Tags::Phi< Dim > > > * > evolved_fields, 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)
	Reconstruct the fields \((\Pi, \Phi_i)\) from the characteristic fields.

Detailed Description

Items related to evolving the second-order scalar wave equations.

\begin{align*}\partial_t \Psi &= -\Pi \\ \partial_t \Pi &= -\delta^{ij}\partial_i \partial_j \Psi \end{align*}

Function Documentation

characteristic_fields() [1/2]

template<size_t Dim>

Variables< tmpl::list< Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > > SecondOrderScalarWave::characteristic_fields	(	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 for the second-order scalar wave system.

The characteristic fields are given in terms of the evolved fields by:

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

where \(n_i\) is the unit normal along which the characteristic fields are defined. Note that, unlike the first-order scalar wave system, there is no characteristic field corresponding to \(\Psi\) (see [92]).

The corresponding characteristic speeds are computed by Tags::CharacteristicSpeedsCompute . The inverse transform, reconstructing \((\Pi, \Phi_i)\) from the characteristic fields, is computed by Tags::FieldsFromInverseCharacteristicTransformCompute .

characteristic_fields() [2/2]

template<size_t Dim>

void SecondOrderScalarWave::characteristic_fields	(	gsl::not_null< Variables< tmpl::list< Tags::VZero< Dim >, Tags::VPlus, Tags::VMinus > > * >	char_fields,
		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 for the second-order scalar wave system.

The characteristic fields are given in terms of the evolved fields by:

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

where \(n_i\) is the unit normal along which the characteristic fields are defined. Note that, unlike the first-order scalar wave system, there is no characteristic field corresponding to \(\Psi\) (see [92]).

The corresponding characteristic speeds are computed by Tags::CharacteristicSpeedsCompute . The inverse transform, reconstructing \((\Pi, \Phi_i)\) from the characteristic fields, is computed by Tags::FieldsFromInverseCharacteristicTransformCompute .

characteristic_speeds() [1/2]

template<size_t Dim>

std::array< DataVector, 3 > SecondOrderScalarWave::characteristic_speeds

(

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

unit_normal_one_form

)

Compute the characteristic speeds for the second-order scalar wave system.

The characteristic fields are, in order, \(v^0_i\) (Tags::VZero), \(v^+\) (Tags::VPlus), and \(v^-\) (Tags::VMinus). Their characteristic speeds are 0, +1, and -1, respectively.

characteristic_speeds() [2/2]

template<size_t Dim>

void SecondOrderScalarWave::characteristic_speeds	(	gsl::not_null< std::array< DataVector, 3 > * >	char_speeds,
		const tnsr::i< DataVector, Dim, Frame::Inertial > &	unit_normal_one_form )

Compute the characteristic speeds for the second-order scalar wave system.

The characteristic fields are, in order, \(v^0_i\) (Tags::VZero), \(v^+\) (Tags::VPlus), and \(v^-\) (Tags::VMinus). Their characteristic speeds are 0, +1, and -1, respectively.

fields_from_inverse_characteristic_transform() [1/2]

template<size_t Dim>

Variables< tmpl::list< Tags::Pi, Tags::Phi< Dim > > > SecondOrderScalarWave::fields_from_inverse_characteristic_transform	(	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 )

Reconstruct the fields \((\Pi, \Phi_i)\) from the characteristic fields.

This uses the inverse of the relations in Tags::CharacteristicFieldsCompute :

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

The scalar field \(\Psi\) is not reconstructed because it is not part of the characteristic decomposition of the second-order scalar wave system.

fields_from_inverse_characteristic_transform() [2/2]

template<size_t Dim>

void SecondOrderScalarWave::fields_from_inverse_characteristic_transform	(	gsl::not_null< Variables< tmpl::list< Tags::Pi, Tags::Phi< Dim > > > * >	evolved_fields,
		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 )

Reconstruct the fields \((\Pi, \Phi_i)\) from the characteristic fields.

This uses the inverse of the relations in Tags::CharacteristicFieldsCompute :

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

The scalar field \(\Psi\) is not reconstructed because it is not part of the characteristic decomposition of the second-order scalar wave system.