naginterfaces.library.pde.dim1_​parab_​euler_​exact

naginterfaces.library.pde.dim1_parab_euler_exact(uleft, uright, gamma, tol, niter, comm)

dim1_parab_euler_exact calculates a numerical flux function using an Exact Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes dim1_parab_convdiff(), dim1_parab_convdiff_dae() or dim1_parab_convdiff_remesh(), but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

For full information please refer to the NAG Library document for d03px

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d03/d03pxf.html

Parameters

uleftfloat, array-like, shape $\left(3\right)$

$uleft[\text{i}-1]$ must contain the left value of the component $U_{\text{i}}$, for $\text{i}=1,2,\dots ,3$. That is, $uleft\left[0\right]$ must contain the left value of $\rho$, $uleft\left[1\right]$ must contain the left value of $m$ and $uleft\left[2\right]$ must contain the left value of $e$.

urightfloat, array-like, shape $\left(3\right)$

$uright[\text{i}-1]$ must contain the right value of the component $U_{\text{i}}$, for $\text{i}=1,2,\dots ,3$. That is, $uright\left[0\right]$ must contain the right value of $\rho$, $uright\left[1\right]$ must contain the right value of $m$ and $uright\left[2\right]$ must contain the right value of $e$.

gammafloat

The ratio of specific heats, $\gamma$.

tolfloat

The tolerance to be used in the Newton–Raphson procedure to calculate the pressure. If $tol$ is set to zero then the default value of $1.0\times {10}^{-6}$ is used.

niterint

The maximum number of Newton–Raphson iterations allowed. If $niter$ is set to zero then the default value of $20$ is used.

commdict, communication object

Communication structure.

On initial entry: need not be set.

Returns

fluxfloat, ndarray, shape $\left(3\right)$

$flux[\text{i}-1]$ contains the numerical flux component ${\hat{F}}_{\text{i}}$, for $\text{i}=1,2,\dots ,3$.

Raises

NagValueError

(errno $1$)

On entry, $niter=⟨value⟩$.

Constraint: $niter\ge 0$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol\ge 0.0$.

(errno $1$)

On entry, $gamma=⟨value⟩$.

Constraint: $gamma>0.0$.

(errno $2$)

Right pressure value $\text{pr}<0.0$: $\text{pr}=⟨value⟩$.

(errno $2$)

Left pressure value $\text{pl}<0.0$: $\text{pl}=⟨value⟩$.

(errno $2$)

On entry, $uright\left[0\right]=⟨value⟩$.

Constraint: $uright\left[0\right]\ge 0.0$.

(errno $2$)

On entry, $uleft\left[0\right]=⟨value⟩$.

Constraint: $uleft\left[0\right]\ge 0.0$.

(errno $3$)

A vacuum condition has been detected.

(errno $4$)

Newton–Raphson iteration failed to converge.

Notes

dim1_parab_euler_exact calculates a numerical flux function at a single spatial point using an Exact Riemann Solver (see Toro (1996) and Toro (1989)) for the Euler equations (for a perfect gas) in conservative form. You must supply the left and right solution values at the point where the numerical flux is required, i.e., the initial left and right states of the Riemann problem defined below. In dim1_parab_convdiff(), dim1_parab_convdiff_dae() and dim1_parab_convdiff_remesh(), the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the function argument $\text{numflx}$ from which you may call dim1_parab_euler_exact.

The Euler equations for a perfect gas in conservative form are:

$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}=0\text{,}$$

with

$$\begin{array}{c}U=\left[\begin{array}{c}\rho \\ m\\ e\end{array}\right]\text{and}F=\left[\begin{array}{c}m\\ \frac{m^2}{\rho }+(\gamma -1)(e-\frac{m^2}{2\rho })\\ \frac{me}{\rho }+\frac{m}{\rho }(\gamma -1)(e-\frac{m^2}{2\rho })\end{array}\right]\text{,}\end{array}$$

where $\rho$ is the density, $m$ is the momentum, $e$ is the specific total energy and $\gamma$ is the (constant) ratio of specific heats. The pressure $p$ is given by

$$p=(\gamma -1)(e-\frac{\rho u^2}{2})\text{,}$$

where $u=m/\rho$ is the velocity.

The function calculates the numerical flux function $F(U_L,U_R)=F\left(U^{\ast }(U_L,U_R)\right)$, where $U=U_L$ and $U=U_R$ are the left and right solution values, and $U^{\ast }(U_L,U_R)$ is the intermediate state $\omega \left(0\right)$ arising from the similarity solution $U(y,t)=\omega \left(y/t\right)$ of the Riemann problem defined by

$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial y}=0\text{,}$$

with $U$ and $F$ as in (2), and initial piecewise constant values $U=U_L$ for $y<0$ and $U=U_R$ for $y>0$. The spatial domain is $-\infty <y<\infty$, where $y=0$ is the point at which the numerical flux is required.

The algorithm is termed an Exact Riemann Solver although it does in fact calculate an approximate solution to a true Riemann problem, as opposed to an Approximate Riemann Solver which involves some form of alternative modelling of the Riemann problem. The approximation part of the Exact Riemann Solver is a Newton–Raphson iterative procedure to calculate the pressure, and you must supply a tolerance $tol$ and a maximum number of iterations $niter$. Default values for these arguments can be chosen.

A solution cannot be found by this function if there is a vacuum state in the Riemann problem (loosely characterised by zero density), or if such a state is generated by the interaction of two non-vacuum data states. In this case a Riemann solver which can handle vacuum states has to be used (see Toro (1996)).

References

Toro, E F, 1989, A weighted average flux method for hyperbolic conservation laws, Proc. Roy. Soc. Lond. (A423), 401–418

Toro, E F, 1996, Riemann Solvers and Upwind Methods for Fluid Dynamics, Springer–Verlag