NewtonianEuler Namespace Reference

Items related to evolving the Newtonian Euler system. More...

Namespaces
namespace	AnalyticData
	Holds classes implementing analytic data for the NewtonianEuler system.
namespace	BoundaryConditions
	Boundary conditions for the Newtonian Euler hydrodynamics system.
namespace	BoundaryCorrections
	Boundary corrections/numerical fluxes.
namespace	fd
	Finite difference functionality for Newtonian Euler.
namespace	OptionTags
	OptionTags for the conservative formulation of the Newtonian Euler system
namespace	Solutions
	Holds classes implementing a solution to the Newtonian Euler system.
namespace	Sources
namespace	subcell
	Code required by the DG-subcell/FD hybrid solver.
namespace	Tags
	Tags for the conservative formulation of the Newtonian Euler system

Classes
struct	ComputeFluxes
	Compute the fluxes of the conservative variables of the Newtonian Euler system. More...
struct	ConservativeFromPrimitive
	Compute the conservative variables from the primitive variables. More...
struct	PrimitiveFromConservative
	Compute the primitive variables from the conservative variables. More...
struct	System
struct	TimeDerivativeTerms
	Compute the time derivative of the conserved variables for the Newtonian Euler system. More...

Functions
template<size_t Dim>
std::pair< DataVector, std::pair< Matrix, Matrix > >	numerical_eigensystem (const tnsr::I< double, Dim > &velocity, const Scalar< double > &sound_speed_squared, const Scalar< double > &specific_enthalpy, const Scalar< double > &kappa_over_density, const tnsr::i< double, Dim > &unit_normal)
	Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system. More...
template<size_t Dim>
void	characteristic_speeds (gsl::not_null< std::array< DataVector, Dim+2 > * > char_speeds, const tnsr::I< DataVector, Dim > &velocity, const Scalar< DataVector > &sound_speed, const tnsr::i< DataVector, Dim > &normal)
	Compute the characteristic speeds of NewtonianEuler system. More...
template<size_t Dim>
std::array< DataVector, Dim+2 >	characteristic_speeds (const tnsr::I< DataVector, Dim > &velocity, const Scalar< DataVector > &sound_speed, const tnsr::i< DataVector, Dim > &normal)
	Compute the characteristic speeds of NewtonianEuler system. More...
template<size_t Dim>
Matrix	right_eigenvectors (const tnsr::I< double, Dim > &velocity, const Scalar< double > &sound_speed_squared, const Scalar< double > &specific_enthalpy, const Scalar< double > &kappa_over_density, const tnsr::i< double, Dim > &unit_normal)
	Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system. More...
template<size_t Dim>
Matrix	left_eigenvectors (const tnsr::I< double, Dim > &velocity, const Scalar< double > &sound_speed_squared, const Scalar< double > &specific_enthalpy, const Scalar< double > &kappa_over_density, const tnsr::i< double, Dim > &unit_normal)
	Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system. More...
template<typename DataType >
void	internal_energy_density (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &mass_density, const Scalar< DataType > &specific_internal_energy)
template<typename DataType >
Scalar< DataType >	internal_energy_density (const Scalar< DataType > &mass_density, const Scalar< DataType > &specific_internal_energy)
template<typename DataType , size_t Dim, typename Fr >
void	kinetic_energy_density (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &mass_density, const tnsr::I< DataType, Dim, Fr > &velocity)
template<typename DataType , size_t Dim, typename Fr >
Scalar< DataType >	kinetic_energy_density (const Scalar< DataType > &mass_density, const tnsr::I< DataType, Dim, Fr > &velocity)
template<typename DataType , size_t Dim, typename Fr >
void	mach_number (gsl::not_null< Scalar< DataType > * > result, const tnsr::I< DataType, Dim, Fr > &velocity, const Scalar< DataType > &sound_speed)
template<typename DataType , size_t Dim, typename Fr >
Scalar< DataType >	mach_number (const tnsr::I< DataType, Dim, Fr > &velocity, const Scalar< DataType > &sound_speed)
template<typename DataType , size_t Dim, typename Fr >
void	ram_pressure (gsl::not_null< tnsr::II< DataType, Dim, Fr > * > result, const Scalar< DataType > &mass_density, const tnsr::I< DataType, Dim, Fr > &velocity)
template<typename DataType , size_t Dim, typename Fr >
tnsr::II< DataType, Dim, Fr >	ram_pressure (const Scalar< DataType > &mass_density, const tnsr::I< DataType, Dim, Fr > &velocity)
template<typename DataType , size_t ThermodynamicDim>
void	sound_speed_squared (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &mass_density, const Scalar< DataType > &specific_internal_energy, const EquationsOfState::EquationOfState< false, ThermodynamicDim > &equation_of_state)
template<typename DataType , size_t ThermodynamicDim>
Scalar< DataType >	sound_speed_squared (const Scalar< DataType > &mass_density, const Scalar< DataType > &specific_internal_energy, const EquationsOfState::EquationOfState< false, ThermodynamicDim > &equation_of_state)
template<typename DataType , size_t Dim, typename Fr >
void	specific_kinetic_energy (gsl::not_null< Scalar< DataType > * > result, const tnsr::I< DataType, Dim, Fr > &velocity)
template<typename DataType , size_t Dim, typename Fr >
Scalar< DataType >	specific_kinetic_energy (const tnsr::I< DataType, Dim, Fr > &velocity)

Detailed Description

Items related to evolving the Newtonian Euler system.

Function Documentation

characteristic_speeds() [1/2]

template<size_t Dim>

std::array< DataVector, Dim+2 > NewtonianEuler::characteristic_speeds	(	const tnsr::I< DataVector, Dim > &	velocity,
		const Scalar< DataVector > &	sound_speed,
		const tnsr::i< DataVector, Dim > &	normal
	)

Compute the characteristic speeds of NewtonianEuler system.

The principal symbol of the system is diagonalized so that the elements of the diagonal matrix are the characteristic speeds

$$\begin{array}{rl}{\lambda }_1& =v_n-c_s,\\ {\lambda }_{i+1}& =v_n,\\ {\lambda }_{\text{Dim}+2}& =v_n+c_s,\end{array}$$

where $i=1,...,\text{Dim}$, $v_n=n_i{v}^i$ is the velocity projected onto the normal, and $c_s$ is the sound speed.

characteristic_speeds() [2/2]

template<size_t Dim>

void NewtonianEuler::characteristic_speeds	(	gsl::not_null< std::array< DataVector, Dim+2 > * >	char_speeds,
		const tnsr::I< DataVector, Dim > &	velocity,
		const Scalar< DataVector > &	sound_speed,
		const tnsr::i< DataVector, Dim > &	normal
	)

Compute the characteristic speeds of NewtonianEuler system.

The principal symbol of the system is diagonalized so that the elements of the diagonal matrix are the characteristic speeds

$$\begin{array}{rl}{\lambda }_1& =v_n-c_s,\\ {\lambda }_{i+1}& =v_n,\\ {\lambda }_{\text{Dim}+2}& =v_n+c_s,\end{array}$$

where $i=1,...,\text{Dim}$, $v_n=n_i{v}^i$ is the velocity projected onto the normal, and $c_s$ is the sound speed.

internal_energy_density() [1/2]

template<typename DataType >

Scalar< DataType > NewtonianEuler::internal_energy_density	(	const Scalar< DataType > &	mass_density,
		const Scalar< DataType > &	specific_internal_energy
	)

Compute the internal energy density, $\rho ϵ$, where $\rho$ is the mass density, and $ϵ$ is the specific internal energy.

internal_energy_density() [2/2]

template<typename DataType >

void NewtonianEuler::internal_energy_density	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	mass_density,
		const Scalar< DataType > &	specific_internal_energy
	)

Compute the internal energy density, $\rho ϵ$, where $\rho$ is the mass density, and $ϵ$ is the specific internal energy.

kinetic_energy_density() [1/2]

template<typename DataType , size_t Dim, typename Fr >

Scalar< DataType > NewtonianEuler::kinetic_energy_density	(	const Scalar< DataType > &	mass_density,
		const tnsr::I< DataType, Dim, Fr > &	velocity
	)

Compute the kinetic energy density, $\rho v^2/2$, where $\rho$ is the mass density, and $v$ is the magnitude of the velocity.

kinetic_energy_density() [2/2]

template<typename DataType , size_t Dim, typename Fr >

void NewtonianEuler::kinetic_energy_density	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	mass_density,
		const tnsr::I< DataType, Dim, Fr > &	velocity
	)

Compute the kinetic energy density, $\rho v^2/2$, where $\rho$ is the mass density, and $v$ is the magnitude of the velocity.

left_eigenvectors()

template<size_t Dim>

Matrix NewtonianEuler::left_eigenvectors	(	const tnsr::I< double, Dim > &	velocity,
		const Scalar< double > &	sound_speed_squared,
		const Scalar< double > &	specific_enthalpy,
		const Scalar< double > &	kappa_over_density,
		const tnsr::i< double, Dim > &	unit_normal
	)

Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system.

Let $u$ be the conserved (i.e., evolved) variables of the Newtonian Euler system, and $w$ the characteristic variables of this system with respect to a unit normal one form $n_i$. The function left_eigenvectors computes the matrix ${\mathrm{\Omega }}_L$ corresponding to the transform $w={\mathrm{\Omega }}_{L}u$. The function right_eigenvectors computes the matrix ${\mathrm{\Omega }}_R$ corresponding to the inverse transform $u={\mathrm{\Omega }}_{R}w$. Here the components of $u$ are ordered as $u=\{\rho ,\rho v_x,\rho v_y,\rho v_z,e\}$ in 3D, and the components of $w$ are ordered by their corresponding eigenvalues (i.e., characteristic speeds) $\lambda =\{v_n-c_s,v_n,v_n,v_n,v_n+c_s\}$. In these expressions, $\rho$ is the fluid mass density, $v_{x,y,z}$ are the components of the fluid velocity, $e$ is the total energy density, $v_n$ is the component of the velocity along the unit normal $n_i$, and $c_s$ is the sound speed.

For a short discussion of the characteristic transformation and the matrices ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$, see [122] Chapter 3.

Here we briefly summarize the procedure. With $F^x(u)$ the Newtonian Euler flux in direction $x$, then the flux Jacobian along $x$ is the matrix $A_x=\partial F_{\beta }^x(u)/\partial u_{\alpha }$. The indices $\alpha ,\beta$ range over the different evolved fields. In higher dimensions, the flux Jacobian along the unit normal $n_i$ is $A=n_x{A}_x+n_y{A}_y+n_z{A}_z$. This matrix can be diagonalized as $A={\mathrm{\Omega }}_R\mathrm{\Lambda }{\mathrm{\Omega }}_L$. Here $\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(v_n-c_s,v_n,v_n,v_n,v_n+c_s)$ is a diagonal matrix containing the characteristic speeds; ${\mathrm{\Omega }}_R$ is a matrix whose columns are the right eigenvectors of $A$; ${\mathrm{\Omega }}_L$ is the inverse of $R$.

mach_number() [1/2]

template<typename DataType , size_t Dim, typename Fr >

Scalar< DataType > NewtonianEuler::mach_number	(	const tnsr::I< DataType, Dim, Fr > &	velocity,
		const Scalar< DataType > &	sound_speed
	)

Compute the local Mach number, $\text{Ma}=v/c_s$, where $v$ is the magnitude of the velocity, and $c_s$ is the sound speed.

mach_number() [2/2]

template<typename DataType , size_t Dim, typename Fr >

void NewtonianEuler::mach_number	(	gsl::not_null< Scalar< DataType > * >	result,
		const tnsr::I< DataType, Dim, Fr > &	velocity,
		const Scalar< DataType > &	sound_speed
	)

Compute the local Mach number, $\text{Ma}=v/c_s$, where $v$ is the magnitude of the velocity, and $c_s$ is the sound speed.

numerical_eigensystem()

template<size_t Dim>

std::pair< DataVector, std::pair< Matrix, Matrix > > NewtonianEuler::numerical_eigensystem	(	const tnsr::I< double, Dim > &	velocity,
		const Scalar< double > &	sound_speed_squared,
		const Scalar< double > &	specific_enthalpy,
		const Scalar< double > &	kappa_over_density,
		const tnsr::i< double, Dim > &	unit_normal
	)

Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system.

See right_eigenvectors and left_eigenvectors for more details.

However, note that this function computes the transformation (i.e., the eigenvectors of the flux Jacobian) numerically, instead of using the analytic expressions. This is useful as a proof-of-concept for more complicated systems where the analytic expressions may not be known.

ram_pressure() [1/2]

template<typename DataType , size_t Dim, typename Fr >

tnsr::II< DataType, Dim, Fr > NewtonianEuler::ram_pressure	(	const Scalar< DataType > &	mass_density,
		const tnsr::I< DataType, Dim, Fr > &	velocity
	)

Compute the ram pressure, $\rho v^i{v}^j$, where $\rho$ is the mass density, and $v^i$ is the velocity.

ram_pressure() [2/2]

template<typename DataType , size_t Dim, typename Fr >

void NewtonianEuler::ram_pressure	(	gsl::not_null< tnsr::II< DataType, Dim, Fr > * >	result,
		const Scalar< DataType > &	mass_density,
		const tnsr::I< DataType, Dim, Fr > &	velocity
	)

Compute the ram pressure, $\rho v^i{v}^j$, where $\rho$ is the mass density, and $v^i$ is the velocity.

right_eigenvectors()

template<size_t Dim>

Matrix NewtonianEuler::right_eigenvectors	(	const tnsr::I< double, Dim > &	velocity,
		const Scalar< double > &	sound_speed_squared,
		const Scalar< double > &	specific_enthalpy,
		const Scalar< double > &	kappa_over_density,
		const tnsr::i< double, Dim > &	unit_normal
	)

Compute the transform matrices between the conserved variables and the characteristic variables of the NewtonianEuler system.

Let $u$ be the conserved (i.e., evolved) variables of the Newtonian Euler system, and $w$ the characteristic variables of this system with respect to a unit normal one form $n_i$. The function left_eigenvectors computes the matrix ${\mathrm{\Omega }}_L$ corresponding to the transform $w={\mathrm{\Omega }}_{L}u$. The function right_eigenvectors computes the matrix ${\mathrm{\Omega }}_R$ corresponding to the inverse transform $u={\mathrm{\Omega }}_{R}w$. Here the components of $u$ are ordered as $u=\{\rho ,\rho v_x,\rho v_y,\rho v_z,e\}$ in 3D, and the components of $w$ are ordered by their corresponding eigenvalues (i.e., characteristic speeds) $\lambda =\{v_n-c_s,v_n,v_n,v_n,v_n+c_s\}$. In these expressions, $\rho$ is the fluid mass density, $v_{x,y,z}$ are the components of the fluid velocity, $e$ is the total energy density, $v_n$ is the component of the velocity along the unit normal $n_i$, and $c_s$ is the sound speed.

For a short discussion of the characteristic transformation and the matrices ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$, see [122] Chapter 3.

Here we briefly summarize the procedure. With $F^x(u)$ the Newtonian Euler flux in direction $x$, then the flux Jacobian along $x$ is the matrix $A_x=\partial F_{\beta }^x(u)/\partial u_{\alpha }$. The indices $\alpha ,\beta$ range over the different evolved fields. In higher dimensions, the flux Jacobian along the unit normal $n_i$ is $A=n_x{A}_x+n_y{A}_y+n_z{A}_z$. This matrix can be diagonalized as $A={\mathrm{\Omega }}_R\mathrm{\Lambda }{\mathrm{\Omega }}_L$. Here $\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(v_n-c_s,v_n,v_n,v_n,v_n+c_s)$ is a diagonal matrix containing the characteristic speeds; ${\mathrm{\Omega }}_R$ is a matrix whose columns are the right eigenvectors of $A$; ${\mathrm{\Omega }}_L$ is the inverse of $R$.

sound_speed_squared() [1/2]

template<typename DataType , size_t ThermodynamicDim>

Scalar< DataType > NewtonianEuler::sound_speed_squared	(	const Scalar< DataType > &	mass_density,
		const Scalar< DataType > &	specific_internal_energy,
		const EquationsOfState::EquationOfState< false, ThermodynamicDim > &	equation_of_state
	)

Compute the Newtonian sound speed squared $c_s^2=\chi +p\kappa /{\rho }^2$, where $p$ is the fluid pressure, $\rho$ is the mass density, $\chi =(\partial p/\partial \rho {)}_{ϵ}$ and $\kappa =(\partial p/\partial ϵ{)}_{\rho }$, where $ϵ$ is the specific internal energy.

sound_speed_squared() [2/2]

template<typename DataType , size_t ThermodynamicDim>

void NewtonianEuler::sound_speed_squared	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	mass_density,
		const Scalar< DataType > &	specific_internal_energy,
		const EquationsOfState::EquationOfState< false, ThermodynamicDim > &	equation_of_state
	)

Compute the Newtonian sound speed squared $c_s^2=\chi +p\kappa /{\rho }^2$, where $p$ is the fluid pressure, $\rho$ is the mass density, $\chi =(\partial p/\partial \rho {)}_{ϵ}$ and $\kappa =(\partial p/\partial ϵ{)}_{\rho }$, where $ϵ$ is the specific internal energy.

specific_kinetic_energy() [1/2]

template<typename DataType , size_t Dim, typename Fr >

Scalar< DataType > NewtonianEuler::specific_kinetic_energy

(

const tnsr::I< DataType, Dim, Fr > &

velocity

)

Compute the specific kinetic energy, $v^2/2$, where $v$ is the magnitude of the velocity.

specific_kinetic_energy() [2/2]

template<typename DataType , size_t Dim, typename Fr >

void NewtonianEuler::specific_kinetic_energy	(	gsl::not_null< Scalar< DataType > * >	result,
		const tnsr::I< DataType, Dim, Fr > &	velocity
	)

Compute the specific kinetic energy, $v^2/2$, where $v$ is the magnitude of the velocity.