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.

2 Specification

#include <nag.h>

void	d03pvc (const double uleft[], const double uright[], double gamma, Nag_OsherVersion path, double flux[], Nag_D03_Save saved, NagError fail)

The function may be called by the names: d03pvc, nag_pde_dim1_parab_euler_osher or nag_pde_parab_1d_euler_osher.

3 Description

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:

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

(1)

with

U = [\begin{matrix} ρ \\ m \\ e \end{matrix}] and F = [\begin{matrix} m \\ \frac{m^{2}}{ρ} + (γ - 1) (e - \frac{m^{2}}{2 ρ}) \\ \frac{m e}{ρ} + \frac{m}{ρ} (γ - 1) (e - \frac{m^{2}}{2 ρ}) \end{matrix}],

(2)

where

ρ

is the density,

m

is the momentum,

e

is the specific total energy, and

γ

is the (constant) ratio of specific heats. The pressure

p

is given by

p = (γ - 1) (e - \frac{ρ u^{2}}{2}),

(3)

where

u = m / ρ

is the velocity.

The function calculates the Osher approximation to the numerical flux function

F (U_{L}, U_{R}) = F (U^{*} (U_{L}, U_{R}))

, 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

ω (0)

arising from the similarity solution

U (y, t) = ω (y / t)

of the Riemann problem defined by

\frac{\partial U}{\partial t} + \frac{\partial F}{\partial 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 < 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).

4 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

5 Arguments

1: $uleft [3]$ – const double Input

On entry:

uleft [i - 1]

must contain the left value of the component

U_{i}

, for

i = 1, 2, 3

. That is,

uleft [0]

must contain the left value of

ρ

uleft [1]

must contain the left value of

m

and

uleft [2]

must contain the left value of

e

Constraints:

$uleft [0] \geq 0.0$ ;
Left pressure, $pl \geq 0.0$ , where $pl$ is calculated using (3).

2: $uright [3]$ – const double Input

On entry:

uright [i - 1]

must contain the right value of the component

U_{i}

, for

i = 1, 2, 3

. That is,

uright [0]

must contain the right value of

ρ

uright [1]

must contain the right value of

m

and

uright [2]

must contain the right value of

e

Constraints:

$uright [0] \geq 0.0$ ;
Right pressure, $pr \geq 0.0$ , where $pr$ is calculated using (3).

3: $gamma$ – double Input

On entry: the ratio of specific heats,

γ

Constraint:

gamma > 0.0

4: $path$ – Nag_OsherVersion Input

On entry: the variant of the Osher scheme.

$path = Nag_OsherOriginal$: Original.
$path = Nag_OsherPhysical$: Physical.

Constraint:

path = Nag_OsherOriginal

Nag_OsherPhysical

5: $flux [3]$ – double Output

On exit:

flux [i - 1]

contains the numerical flux component

{\hat{F}}_{i}

, for

i = 1, 2, 3

6: $saved$ – 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: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: Left pressure value $pl < 0.0$ : $pl = ⟨ value ⟩$ .

On entry, $gamma = ⟨ value ⟩$ .
Constraint: $gamma > 0.0$ .

On entry, $uleft [0] = ⟨ value ⟩$ .
Constraint: $uleft [0] \geq 0.0$ .

On entry, $uright [0] = ⟨ value ⟩$ .
Constraint: $uright [0] \geq 0.0$ .

Right pressure value $pr < 0.0$ : $pr = ⟨ value ⟩$ .

7 Accuracy

d03pvc performs an exact calculation of the Osher numerical flux function, and so the result will be accurate to machine precision.

8 Parallelism and Performance

d03pvc is not threaded in any implementation.

9 Further Comments

d03pvc must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with

uleft [i - 1]

and

uright [i - 1]

containing the left and right values of

ρ, m

and

e

, for

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.