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

naginterfaces.library.pde.dim1_parab_euler_osher(uleft, uright, gamma, path, comm)

dim1_parab_euler_osher calculates a numerical flux function using Osher’s Approximate 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 d03pv

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d03/d03pvf.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, $\gamma$.

pathstr, length 1

The variant of the Osher scheme.

$path=\text{'O'}$

Original.

$path=\text{'P'}$

Physical.

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, $gamma=⟨value⟩$.

Constraint: $gamma>0.0$.

(errno $1$)

On entry, $path=⟨value⟩$.

Constraint: $path=\text{'O'}$ or $\text{'P'}$.

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

Notes

dim1_parab_euler_osher calculates a numerical flux function at a single spatial point using Osher’s Approximate Riemann Solver (see Hemker and Spekreijse (1986) and Pennington and Berzins (1994)) 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 the functions 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_osher.

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 Osher approximation to 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. Osher’s solver carries out an integration along a path in the phase space of $U$ consisting of subpaths which are piecewise parallel to the eigenvectors of the Jacobian of the PDE system. There are two variants of the Osher solver termed O (original) and P (physical), which differ in the order in which the subpaths are taken. The P-variant is generally more efficient, but in some rare cases may fail (see Hemker and Spekreijse (1986) for details). The argument $path$ specifies which variant is to be used. The algorithm for Osher’s solver for the Euler equations is given in detail in the Appendix of Pennington and Berzins (1994).

References

Hemker, P W and Spekreijse, S P, 1986, Multiple grid and Osher’s scheme for the efficient solution of the steady Euler equations, Applied Numerical Mathematics (2), 475–493

Pennington, S V and Berzins, M, 1994, New NAG Library software for first-order partial differential equations, ACM Trans. Math. Softw. (20), 63–99

Quirk, J J, 1994, A contribution to the great Riemann solver debate, Internat. J. Numer. Methods Fluids (18), 555–574