hydro::initial_data::irrotational_bns Namespace Reference

Items related to solving for irrotational bns initial data See e.g. [13] Ch. 15 (P. 523) More...

Functions
template<typename DataType >
void	rotational_shift (gsl::not_null< tnsr::I< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &shift, const tnsr::I< DataType, 3 > &spatial_rotational_killing_vector)
	Compute the shift plus a spatial vector \( k^i\) representing the local binary rotation \(B^i = \beta^i + k^i\).
template<typename DataType >
tnsr::I< DataType, 3 >	rotational_shift (const tnsr::I< DataType, 3 > &shift, const tnsr::I< DataType, 3 > &spatial_rotational_killing_vector)
	Compute the shift plus a spatial vector \( k^i\) representing the local binary rotation \(B^i = \beta^i + k^i\).
template<typename DataType >
void	rotational_shift_stress (gsl::not_null< tnsr::II< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &rotational_shift, const Scalar< DataType > &lapse)
	Compute the stress-energy corresponding to the rotation shift. (this has no corresponding equation number in [13], it is defined for convenience in evaluating fluxes and sources for the DG scheme.) More...
template<typename DataType >
tnsr::II< DataType, 3 >	rotational_shift_stress (const tnsr::I< DataType, 3 > &rotational_shift, const Scalar< DataType > &lapse)
	Compute the stress-energy corresponding to the rotation shift. (this has no corresponding equation number in [13], it is defined for convenience in evaluating fluxes and sources for the DG scheme.) More...
template<typename DataType >
void	derivative_rotational_shift_over_lapse_squared (gsl::not_null< tnsr::iJ< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &rotational_shift, const tnsr::iJ< DataType, 3 > &deriv_of_shift, const Scalar< DataType > &lapse, const tnsr::i< DataType, 3 > &deriv_of_lapse, const tnsr::iJ< DataType, 3 > &deriv_of_spatial_rotational_killing_vector)
	Compute derivative \( \partial_i (B^j / \alpha^2) \). More...
template<typename DataType >
tnsr::iJ< DataType, 3 >	derivative_rotational_shift_over_lapse_squared (const tnsr::I< DataType, 3 > &rotational_shift, const tnsr::iJ< DataType, 3 > &deriv_of_shift, const Scalar< DataType > &lapse, const tnsr::i< DataType, 3 > &deriv_of_lapse, const tnsr::iJ< DataType, 3 > &deriv_of_spatial_rotational_killing_vector)
	Compute derivative \( \partial_i (B^j / \alpha^2) \). More...
template<typename DataType >
void	specific_enthalpy_squared (gsl::not_null< Scalar< DataType > * > result, const tnsr::I< DataType, 3 > &rotational_shift, const Scalar< DataType > &lapse, const tnsr::i< DataType, 3 > &velocity_potential_gradient, const tnsr::II< DataType, 3 > &inverse_spatial_metric, double euler_enthalpy_constant)
	Compute the specific enthalpy squared from other hydro variables and the spacetime. More...
template<typename DataType >
Scalar< DataType >	specific_enthalpy_squared (const tnsr::I< DataType, 3 > &rotational_shift, const Scalar< DataType > &lapse, const tnsr::i< DataType, 3 > &velocity_potential_gradient, const tnsr::II< DataType, 3 > &inverse_spatial_metric, double euler_enthalpy_constant)
	Compute the specific enthalpy squared from other hydro variables and the spacetime. More...
template<typename DataType >
void	spatial_rotational_killing_vector (gsl::not_null< tnsr::I< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &x, double orbital_angular_velocity, const Scalar< DataType > &sqrt_det_spatial_metric)
	Compute the spatial rotational killing vector associated with uniform rotation around the z-axis. More...
template<typename DataType >
tnsr::I< DataType, 3 >	spatial_rotational_killing_vector (const tnsr::I< DataType, 3 > &x, double orbital_angular_velocity, const Scalar< DataType > &sqrt_det_spatial_metric)
	Compute the spatial rotational killing vector associated with uniform rotation around the z-axis. More...
template<typename DataType >
void	divergence_spatial_rotational_killing_vector (gsl::not_null< Scalar< DataType > * > result, const tnsr::I< DataType, 3 > &x, double orbital_angular_velocity, const Scalar< DataType > &sqrt_det_spatial_metric)
	The spatial derivative of the spatial rotational killing vector. More...
template<typename DataType >
Scalar< DataType >	divergence_spatial_rotational_killing_vector (const tnsr::I< DataType, 3 > &x, double orbital_angular_velocity, const Scalar< DataType > &sqrt_det_spatial_metric)
	The spatial derivative of the spatial rotational killing vector. More...

Detailed Description

Items related to solving for irrotational bns initial data See e.g. [13] Ch. 15 (P. 523)

Function Documentation

derivative_rotational_shift_over_lapse_squared() [1/2]

template<typename DataType >

tnsr::iJ< DataType, 3 > hydro::initial_data::irrotational_bns::derivative_rotational_shift_over_lapse_squared	(	const tnsr::I< DataType, 3 > &	rotational_shift,
		const tnsr::iJ< DataType, 3 > &	deriv_of_shift,
		const Scalar< DataType > &	lapse,
		const tnsr::i< DataType, 3 > &	deriv_of_lapse,
		const tnsr::iJ< DataType, 3 > &	deriv_of_spatial_rotational_killing_vector
	)

Compute derivative \( \partial_i (B^j / \alpha^2) \).

Here \( \partial_i \) is the spatial partial derivative, \( \alpha\) is the lapse and \(B^i\) the rotational shift). The derivatives passed as arguments should be spatial partial derivatives.

derivative_rotational_shift_over_lapse_squared() [2/2]

template<typename DataType >

void hydro::initial_data::irrotational_bns::derivative_rotational_shift_over_lapse_squared	(	gsl::not_null< tnsr::iJ< DataType, 3 > * >	result,
		const tnsr::I< DataType, 3 > &	rotational_shift,
		const tnsr::iJ< DataType, 3 > &	deriv_of_shift,
		const Scalar< DataType > &	lapse,
		const tnsr::i< DataType, 3 > &	deriv_of_lapse,
		const tnsr::iJ< DataType, 3 > &	deriv_of_spatial_rotational_killing_vector
	)

Compute derivative \( \partial_i (B^j / \alpha^2) \).

Here \( \partial_i \) is the spatial partial derivative, \( \alpha\) is the lapse and \(B^i\) the rotational shift). The derivatives passed as arguments should be spatial partial derivatives.

divergence_spatial_rotational_killing_vector() [1/2]

template<typename DataType >

Scalar< DataType > hydro::initial_data::irrotational_bns::divergence_spatial_rotational_killing_vector	(	const tnsr::I< DataType, 3 > &	x,
		double	orbital_angular_velocity,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

The spatial derivative of the spatial rotational killing vector.

As for spatial_rotational_killing_vector, assumes uniform rotation around the z-axis

divergence_spatial_rotational_killing_vector() [2/2]

template<typename DataType >

void hydro::initial_data::irrotational_bns::divergence_spatial_rotational_killing_vector	(	gsl::not_null< Scalar< DataType > * >	result,
		const tnsr::I< DataType, 3 > &	x,
		double	orbital_angular_velocity,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

The spatial derivative of the spatial rotational killing vector.

As for spatial_rotational_killing_vector, assumes uniform rotation around the z-axis

rotational_shift_stress() [1/2]

template<typename DataType >

tnsr::II< DataType, 3 > hydro::initial_data::irrotational_bns::rotational_shift_stress	(	const tnsr::I< DataType, 3 > &	rotational_shift,
		const Scalar< DataType > &	lapse
	)

Compute the stress-energy corresponding to the rotation shift. (this has no corresponding equation number in [13], it is defined for convenience in evaluating fluxes and sources for the DG scheme.)

\[\Sigma^i_j = \frac{1}{2}\frac{B^iB_j}{\alpha^2}\]

rotational_shift_stress() [2/2]

template<typename DataType >

void hydro::initial_data::irrotational_bns::rotational_shift_stress	(	gsl::not_null< tnsr::II< DataType, 3 > * >	result,
		const tnsr::I< DataType, 3 > &	rotational_shift,
		const Scalar< DataType > &	lapse
	)

Compute the stress-energy corresponding to the rotation shift. (this has no corresponding equation number in [13], it is defined for convenience in evaluating fluxes and sources for the DG scheme.)

\[\Sigma^i_j = \frac{1}{2}\frac{B^iB_j}{\alpha^2}\]

spatial_rotational_killing_vector() [1/2]

template<typename DataType >

tnsr::I< DataType, 3 > hydro::initial_data::irrotational_bns::spatial_rotational_killing_vector	(	const tnsr::I< DataType, 3 > &	x,
		double	orbital_angular_velocity,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

Compute the spatial rotational killing vector associated with uniform rotation around the z-axis.

Taking \(\Omega_j\) to be the uniform rotation axis (assumed in the z-direction) and \( \epsilon^{ijk}\) to be the Levi-Civita tensor ( \(\epsilon_{ijk} = \sqrt{\gamma} e_{ijk}\), with \(e_{ijk}\) totally antisymmetric with \( e_{123} = 1\)) , then the killing vector is given by ([13] 15.13) :

\[ k^i = \epsilon^{ijk}\Omega_j x_k \]

spatial_rotational_killing_vector() [2/2]

template<typename DataType >

void hydro::initial_data::irrotational_bns::spatial_rotational_killing_vector	(	gsl::not_null< tnsr::I< DataType, 3 > * >	result,
		const tnsr::I< DataType, 3 > &	x,
		double	orbital_angular_velocity,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

Compute the spatial rotational killing vector associated with uniform rotation around the z-axis.

Taking \(\Omega_j\) to be the uniform rotation axis (assumed in the z-direction) and \( \epsilon^{ijk}\) to be the Levi-Civita tensor ( \(\epsilon_{ijk} = \sqrt{\gamma} e_{ijk}\), with \(e_{ijk}\) totally antisymmetric with \( e_{123} = 1\)) , then the killing vector is given by ([13] 15.13) :

\[ k^i = \epsilon^{ijk}\Omega_j x_k \]

specific_enthalpy_squared() [1/2]

template<typename DataType >

Scalar< DataType > hydro::initial_data::irrotational_bns::specific_enthalpy_squared	(	const tnsr::I< DataType, 3 > &	rotational_shift,
		const Scalar< DataType > &	lapse,
		const tnsr::i< DataType, 3 > &	velocity_potential_gradient,
		const tnsr::II< DataType, 3 > &	inverse_spatial_metric,
		double	euler_enthalpy_constant
	)

Compute the specific enthalpy squared from other hydro variables and the spacetime.

The eqn. is identical in content to [13] 15.76, it computes the specific enthalpy \( h \)

\[ h^2 = \frac{1}{\alpha^2} \left(C + B^i D_i \Phi\right)^2 - D_i \Phi D^i \Phi \]

Where \(\Phi \) is the velocity potential, and \(C\) is the Euler-constant, which in a slowly rotating, slowly orbiting configuration becomes the central specific enthalpy times the central lapse

specific_enthalpy_squared() [2/2]

template<typename DataType >

void hydro::initial_data::irrotational_bns::specific_enthalpy_squared	(	gsl::not_null< Scalar< DataType > * >	result,
		const tnsr::I< DataType, 3 > &	rotational_shift,
		const Scalar< DataType > &	lapse,
		const tnsr::i< DataType, 3 > &	velocity_potential_gradient,
		const tnsr::II< DataType, 3 > &	inverse_spatial_metric,
		double	euler_enthalpy_constant
	)

Compute the specific enthalpy squared from other hydro variables and the spacetime.

The eqn. is identical in content to [13] 15.76, it computes the specific enthalpy \( h \)

\[ h^2 = \frac{1}{\alpha^2} \left(C + B^i D_i \Phi\right)^2 - D_i \Phi D^i \Phi \]

Where \(\Phi \) is the velocity potential, and \(C\) is the Euler-constant, which in a slowly rotating, slowly orbiting configuration becomes the central specific enthalpy times the central lapse