d03pvc 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
d03pfc,
d03plc or
d03psc, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
d03pvc 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
d03pfc,
d03plc and
d03psc, the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the function argument
numflx from which you may call
d03pvc.
The Euler equations for a perfect gas in conservative form are:
with
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
where
$u=m/\rho $ is the velocity.
The function calculates the Osher approximation to the numerical flux function
$F({U}_{L},{U}_{R})=F\left({U}^{*}({U}_{L},{U}_{R})\right)$, where
$U={U}_{L}$ and
$U={U}_{R}$ are the left and right solution values, and
${U}^{*}({U}_{L},{U}_{R})$ is the intermediate state
$\omega \left(0\right)$ arising from the similarity solution
$U(y,t)=\omega (y/t)$ of the Riemann problem defined by
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 Pvariant 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).
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 firstorder partial differential equations ACM Trans. Math. Softw. 20 63–99

1:
$\mathbf{uleft}\left[3\right]$ – const double
Input

On entry: ${\mathbf{uleft}}\left[\mathit{i}1\right]$ must contain the left value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uleft}}\left[0\right]$ must contain the left value of $\rho $, ${\mathbf{uleft}}\left[1\right]$ must contain the left value of $m$ and ${\mathbf{uleft}}\left[2\right]$ must contain the left value of $e$.
Constraints:
 ${\mathbf{uleft}}\left[0\right]\ge 0.0$;
 Left pressure, $\mathit{pl}\ge 0.0$, where $\mathit{pl}$ is calculated using (3).

2:
$\mathbf{uright}\left[3\right]$ – const double
Input

On entry: ${\mathbf{uright}}\left[\mathit{i}1\right]$ must contain the right value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uright}}\left[0\right]$ must contain the right value of $\rho $, ${\mathbf{uright}}\left[1\right]$ must contain the right value of $m$ and ${\mathbf{uright}}\left[2\right]$ must contain the right value of $e$.
Constraints:
 ${\mathbf{uright}}\left[0\right]\ge 0.0$;
 Right pressure, $\mathit{pr}\ge 0.0$, where $\mathit{pr}$ is calculated using (3).

3:
$\mathbf{gamma}$ – double
Input

On entry: the ratio of specific heats, $\gamma $.
Constraint:
${\mathbf{gamma}}>0.0$.

4:
$\mathbf{path}$ – Nag_OsherVersion
Input

On entry: the variant of the Osher scheme.
 ${\mathbf{path}}=\mathrm{Nag\_OsherOriginal}$
 Original.
 ${\mathbf{path}}=\mathrm{Nag\_OsherPhysical}$
 Physical.
Constraint:
${\mathbf{path}}=\mathrm{Nag\_OsherOriginal}$ or $\mathrm{Nag\_OsherPhysical}$.

5:
$\mathbf{flux}\left[3\right]$ – double
Output

On exit: ${\mathbf{flux}}\left[\mathit{i}1\right]$ contains the numerical flux component ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.

6:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

saved may contain data concerning the computation required by
d03pvc as passed through to
numflx from one of the integrator functions
d03pfc,
d03plc or
d03psc. You should not change the components of
saved.

7:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
d03pvc must only be used to calculate the numerical flux for the Euler equations in exactly the form given by
(2), with
${\mathbf{uleft}}\left[\mathit{i}1\right]$ and
${\mathbf{uright}}\left[\mathit{i}1\right]$ containing the left and right values of
$\rho ,m$ and
$e$, for
$\mathit{i}=1,2,3$, respectively. It should be noted that Osher's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see
Quirk (1994) for examples). The time taken depends on the input argument
path and on the left and right solution values, since inclusion of each subpath depends on the signs of the eigenvalues. In general this cannot be determined in advance.