Items related to solving for irrotational bns initial data See e.g. [14] 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 $B^{i} = β^{i} + k^{i}$ .

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 $B^{i} = β^{i} + k^{i}$ .


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 [14], 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 [14], 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 $\partial_{i} (B^{j} / α^{2})$ . 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 $\partial_{i} (B^{j} / α^{2})$ . 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. [14] 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 $\partial_{i} (B^{j} / α^{2})$ .

Here $\partial_{i}$ is the spatial partial derivative, $α$ 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 $\partial_{i} (B^{j} / α^{2})$ .

Here $\partial_{i}$ is the spatial partial derivative, $α$ 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 [14], it is defined for convenience in evaluating fluxes and sources for the DG scheme.)

$Σ^{i j} = \frac{B^{i} B^{j}}{α^{2}}$

◆ 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 [14], it is defined for convenience in evaluating fluxes and sources for the DG scheme.)

$Σ^{i j} = \frac{B^{i} B^{j}}{α^{2}}$

◆ 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 $Ω_{j}$ to be the uniform rotation axis (assumed in the z-direction) and $ϵ^{i j k}$ to be the Levi-Civita tensor ( $ϵ_{i j k} = \sqrt{γ} e_{i j k}$ , with $e_{i j k}$ totally antisymmetric with $e_{123} = 1$ ) , then the killing vector is given by ([14] 15.13) :

$k^{i} = ϵ^{i j k} Ω_{j} x_{k}$

◆ 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 $Ω_{j}$ to be the uniform rotation axis (assumed in the z-direction) and $ϵ^{i j k}$ to be the Levi-Civita tensor ( $ϵ_{i j k} = \sqrt{γ} e_{i j k}$ , with $e_{i j k}$ totally antisymmetric with $e_{123} = 1$ ) , then the killing vector is given by ([14] 15.13) :

$k^{i} = ϵ^{i j k} Ω_{j} x_{k}$

◆ 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 [14] 15.76, it computes the specific enthalpy $h$

$h^{2} = \frac{1}{α^{2}} {(C + B^{i} D_{i} Φ)}^{2} - D_{i} Φ D^{i} Φ$

Where $Φ$ 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 [14] 15.76, it computes the specific enthalpy $h$

$h^{2} = \frac{1}{α^{2}} {(C + B^{i} D_{i} Φ)}^{2} - D_{i} Φ D^{i} Φ$

Where $Φ$ 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

Functions

Detailed Description

Function Documentation

◆ derivative_rotational_shift_over_lapse_squared() [1/2]

◆ derivative_rotational_shift_over_lapse_squared() [2/2]

◆ divergence_spatial_rotational_killing_vector() [1/2]

◆ divergence_spatial_rotational_killing_vector() [2/2]

◆ rotational_shift_stress() [1/2]

◆ rotational_shift_stress() [2/2]

◆ spatial_rotational_killing_vector() [1/2]

◆ spatial_rotational_killing_vector() [2/2]

◆ specific_enthalpy_squared() [1/2]

◆ specific_enthalpy_squared() [2/2]