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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_pde_1d_parab_euler_osher (d03pv)

Purpose

nag_pde_1d_parab_euler_osher (d03pv) 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 nag_pde_1d_parab_convdiff (d03pf), nag_pde_1d_parab_convdiff_dae (d03pl) or nag_pde_1d_parab_convdiff_remesh (d03ps), but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

Syntax

[flux, ifail] = d03pv(uleft, uright, gamma, path)
[flux, ifail] = nag_pde_1d_parab_euler_osher(uleft, uright, gamma, path)

Description

nag_pde_1d_parab_euler_osher (d03pv) 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 nag_pde_1d_parab_convdiff (d03pf), nag_pde_1d_parab_convdiff_dae (d03pl) and nag_pde_1d_parab_convdiff_remesh (d03ps), 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 nag_pde_1d_parab_euler_osher (d03pv).
The Euler equations for a perfect gas in conservative form are:
 $∂U ∂t + ∂F ∂x =0,$ (1)
with
 (2)
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
 $p=γ-1 e-ρu22 ,$ (3)
where $u=m/\rho$ is the velocity.
The function calculates the Osher approximation to the numerical flux function $F\left({U}_{L},{U}_{R}\right)=F\left({U}^{*}\left({U}_{L},{U}_{R}\right)\right)$, where $U={U}_{L}$ and $U={U}_{R}$ are the left and right solution values, and ${U}^{*}\left({U}_{L},{U}_{R}\right)$ is the intermediate state $\omega \left(0\right)$ arising from the similarity solution $U\left(y,t\right)=\omega \left(y/t\right)$ of the Riemann problem defined by
 $∂U ∂t + ∂F ∂y =0,$ (4)
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 , 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

Parameters

Compulsory Input Parameters

1:     $\mathrm{uleft}\left(3\right)$ – double array
${\mathbf{uleft}}\left(\mathit{i}\right)$ must contain the left value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uleft}}\left(1\right)$ must contain the left value of $\rho$, ${\mathbf{uleft}}\left(2\right)$ must contain the left value of $m$ and ${\mathbf{uleft}}\left(3\right)$ must contain the left value of $e$.
Constraints:
• ${\mathbf{uleft}}\left(1\right)\ge 0.0$;
• Left pressure, $\mathit{pl}\ge 0.0$, where $\mathit{pl}$ is calculated using (3).
2:     $\mathrm{uright}\left(3\right)$ – double array
${\mathbf{uright}}\left(\mathit{i}\right)$ must contain the right value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uright}}\left(1\right)$ must contain the right value of $\rho$, ${\mathbf{uright}}\left(2\right)$ must contain the right value of $m$ and ${\mathbf{uright}}\left(3\right)$ must contain the right value of $e$.
Constraints:
• ${\mathbf{uright}}\left(1\right)\ge 0.0$;
• Right pressure, $\mathit{pr}\ge 0.0$, where $\mathit{pr}$ is calculated using (3).
3:     $\mathrm{gamma}$ – double scalar
The ratio of specific heats, $\gamma$.
Constraint: ${\mathbf{gamma}}>0.0$.
4:     $\mathrm{path}$ – string (length ≥ 1)
The variant of the Osher scheme.
${\mathbf{path}}=\text{'O'}$
Original.
${\mathbf{path}}=\text{'P'}$
Physical.
Constraint: ${\mathbf{path}}=\text{'O'}$ or $\text{'P'}$.

None.

Output Parameters

1:     $\mathrm{flux}\left(3\right)$ – double array
${\mathbf{flux}}\left(\mathit{i}\right)$ contains the numerical flux component ${\stackrel{^}{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{gamma}}\le 0.0$, or ${\mathbf{path}}\ne \text{'O'}$ or $\text{'P'}$.
${\mathbf{ifail}}=2$
 On entry, the left and/or right density or pressure value is less than $0.0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

nag_pde_1d_parab_euler_osher (d03pv) performs an exact calculation of the Osher numerical flux function, and so the result will be accurate to machine precision.

nag_pde_1d_parab_euler_osher (d03pv) must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with ${\mathbf{uleft}}\left(\mathit{i}\right)$ and ${\mathbf{uright}}\left(\mathit{i}\right)$ 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.

Example

See Example in nag_pde_1d_parab_convdiff_dae (d03pl).
```function d03pv_example

fprintf('d03pv example results\n\n');

uleft  = [1  0  2.5];
uright = [2  1  0.5];
gamma  = 1.9;
path = 'P';
[flux, ifail] = d03pv(uleft, uright, gamma, path);

fprintf(' Given uleft = [%7.1f %7.1f %7.1f]\n',uleft);
fprintf('  and uright = [%7.1f %7.1f %7.1f]\n',uright);
fprintf('  with gamma =  %7.4f and path = ''%s''\n',gamma,path);
fprintf('\n        flux = [%7.4f %7.4f %7.4f]\n',flux);

```
```d03pv example results

Given uleft = [    1.0     0.0     2.5]
and uright = [    2.0     1.0     0.5]
with gamma =   1.9000 and path = 'P'

flux = [ 0.5540  1.3956  1.9266]
```