nag_pde_1d_parab_euler_roe (d03pu) calculates a numerical flux function using Roe'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] = d03pu(uleft, uright, gamma)

[flux, ifail] = nag_pde_1d_parab_euler_roe(uleft, uright, gamma)

Description

nag_pde_1d_parab_euler_roe (d03pu) calculates a numerical flux function at a single spatial point using Roe's Approximate Riemann Solver (see Roe (1981)) 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_roe (d03pu).

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 Roe 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. This implementation of Roe's scheme for the Euler equations uses the so-called argument-vector method described in Roe (1981).

References

LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag

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

Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372

Parameters

Compulsory Input Parameters

1: $uleft (3)$ – double array

uleft (i)

must contain the left value of the component

U_{i}

, for

i = 1, 2, 3

. That is,

uleft (1)

must contain the left value of

ρ

uleft (2)

must contain the left value of

m

and

uleft (3)

must contain the left value of

e

Constraints:

$uleft (1) \geq 0.0$ ;
Left pressure, $pl \geq 0.0$ , where $pl$ is calculated using (3).

2: $uright (3)$ – double array

uright (i)

must contain the right value of the component

U_{i}

, for

i = 1, 2, 3

. That is,

uright (1)

must contain the right value of

ρ

uright (2)

must contain the right value of

m

and

uright (3)

must contain the right value of

e

Constraints:

$uright (1) \geq 0.0$ ;
Right pressure, $pr \geq 0.0$ , where $pr$ is calculated using (3).

3: $gamma$ – double scalar

The ratio of specific heats,

γ

Constraint:

gamma > 0.0

Optional Input Parameters

None.

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

gamma \leq 0.0

$ifail = 2$

On entry,

the left and/or right density or pressure value is less than

0.0

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_pde_1d_parab_euler_roe (d03pu) performs an exact calculation of the Roe numerical flux function, and so the result will be accurate to machine precision.

Further Comments

nag_pde_1d_parab_euler_roe (d03pu) must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with

uleft (i)

and

uright (i)

containing the left and right values of

ρ, m

and

e

, for

i = 1, 2, 3

, respectively. It should be noted that Roe's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see Quirk (1994) for examples). In particular Roe's scheme does not satisfy an ‘entropy condition’ which guarantees that the approximate solution of the PDE converges to the correct physical solution, and hence it may admit non-physical solutions such as expansion shocks. The algorithm used in this function does not detect or correct any entropy violation. The time taken is independent of the input arguments.

Example

See Example in nag_pde_1d_parab_convdiff_dae (d03pl).

Open in the MATLAB editor: d03pu_example

function d03pu_example


fprintf('d03pu example results\n\n');

uleft  = [1  0  2.5];
uright = [2  1  0.5];
gamma  = 1.9;
[flux, ifail] = d03pu(uleft, uright, gamma);

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\n',gamma);
fprintf('\n        flux = [%7.4f %7.4f %7.4f]\n',flux);

d03pu example results

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

        flux = [ 0.8548  1.3149  1.5161]

PDF version (NAG web site, 64-bit version, 64-bit version)

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