#include "conservation.h"

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Functions
void	flux (const double s, double f, double *e)
	The system is entirely defined by the `flux()` function called by the generic solver for conservation laws.

Variables
scalar	rho []
	The conserved scalars are the gas density \(\rho\) and the total energy \(E\).

scalar	E []

vector	w []

scalar *	scalars = {rho, E}

vector *	vectors = {w}

double	gammao = 1.4

Function Documentation

◆ flux()

void flux	(	const double *	s,
		double *	f,
		double *	e
	)

The system is entirely defined by the flux() function called by the generic solver for conservation laws.

as well as a function which, given the state of each quantity, returns the fluxes and the minimum/maximum eigenvalues (i.e.

The parameter passed to the function is the array s which contains the state variables for each conserved field, in the order of their definition above (i.e. scalars then vectors).

We first recover each value ( \(\rho\), \(E\), \(w_x\) and \(w_y\)) and then compute the corresponding fluxes (f[0], f[1], f[2] and f[3]).

The minimum and maximum eigenvalues for the Euler system are the characteristic speeds \(u \pm \sqrt(\gamma p / \rho)\).

Definition at line 63 of file compressible.h.

References c, dimension, E, f, gammao, i, p, rho, s, sq(), un, and x.

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Variable Documentation

◆ E

scalar E[]

Definition at line 50 of file compressible.h.

Referenced by flux().

◆ gammao

double gammao = 1.4

Definition at line 54 of file compressible.h.

Referenced by flux().

◆ rho

scalar rho[]

The conserved scalars are the gas density \(\rho\) and the total energy \(E\).

Compressible gas dynamics

The Euler system of conservation laws for a compressible gas can be written

\[ \partial_t\left(\begin{array}{c} \rho \\ E \\ w_x \\ w_y \\ \end{array}\right) + \nabla_x \cdot\left(\begin{array}{c} w_x \\ \frac{w_x}{\rho} ( E + p ) \\ \frac{w_x^2}{\rho} + p \\ \frac{w_y w_x}{\rho} \\ \end{array}\right) + \nabla_y \cdot\left(\begin{array}{c} w_y \\ \frac{w_y}{\rho} ( E + p ) \\ \frac{w_y w_x}{\rho} \\ \frac{w_y^2}{\rho} + p \\ \end{array}\right) = 0 \]

with \(\rho\) the gas density, \(E\) the total energy, \(\mathbf{w}\) the gas momentum and \(p\) the pressure given by the equation of state

\[ p = (\gamma - 1)(E - \rho\mathbf{u}^2/2) \]

with \(\gamma\) the polytropic exponent. This system can be solved using the generic solver for systems of conservation laws. The only conserved vector is the momentum \(\mathbf{w}\). The constant \(\gamma\) is represented by gammao here, with a default value of 1.4.

Definition at line 50 of file compressible.h.

Referenced by flux().

◆ scalars

scalar* scalars = {rho, E}

A generic solver for systems of conservation laws

Using the ideas of Kurganov and Tadmor, 2000 it is possible to write a generic solver for systems of conservation laws of the form

\[ \partial_t\left(\begin{array}{c} s_i\\ \mathbf{v}_j\\ \end{array}\right) + \nabla\cdot\left(\begin{array}{c} \mathbf{F}_i\\ \mathbf{T}_j\\ \end{array}\right) = 0 \]

where \(s_i\) is a list of scalar fields, \(\mathbf{v}_j\) a list of vector fields and \(\mathbf{F}_i\), \(\mathbf{T}_j\) are the corresponding vector (resp. tensor) fluxes.

Note that the Saint-Venant solver is a particular case of this generic algorithm.

The user must provide the lists of conserved scalar and vector fields

Definition at line 52 of file compressible.h.

Referenced by event_defaults(), and update_conservation().