NAG Library Routine Document

D03PWF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D03PWF calculates a numerical flux function using a modified HLL (Harten–Lax–van Leer) Approximate Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes D03PFF, D03PLF or D03PSF, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

2 Specification

SUBROUTINE D03PWF (

ULEFT, URIGHT, GAMMA, FLUX, IFAIL)

INTEGER	IFAIL
REAL (KIND=nag_wp)	ULEFT(3), URIGHT(3), GAMMA, FLUX(3)

3 Description

D03PWF calculates a numerical flux function at a single spatial point using a modified HLL (Harten–Lax–van Leer) Approximate Riemann Solver (see Toro (1992), Toro (1996) and Toro et al. (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 D03PFF, D03PLF and D03PSF, the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the subroutine argument NUMFLX from which you may call D03PWF.

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 routine calculates an 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.

4 References

Toro E F (1992) The weighted average flux method applied to the Euler equations Phil. Trans. R. Soc. Lond. A341 499–530

Toro E F (1996) Riemann Solvers and Upwind Methods for Fluid Dynamics Springer–Verlag

Toro E F, Spruce M and Spears W (1994) Restoration of the contact surface in the HLL Riemann solver J. Shock Waves 4 25–34

5 Parameters

1: ULEFT( $3$ ) – REAL (KIND=nag_wp) arrayInput

On entry:

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( $3$ ) – REAL (KIND=nag_wp) arrayInput

On entry:

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 – REAL (KIND=nag_wp)Input

On entry: the ratio of specific heats,

γ

Constraint:

GAMMA > 0.0

4: FLUX( $3$ ) – REAL (KIND=nag_wp) arrayOutput

On exit:

FLUX (i)

contains the numerical flux component

{\hat{F}}_{i}

, for

i = 1, 2, 3

5: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

Note: if the left and/or right values of

ρ

p

(from (3)) are found to be negative, then the routine will terminate with an error exit (

IFAIL = 2

). If the routine is being called from the NUMFLX etc., then a soft fail option (

IFAIL = 1

- 1

) is recommended so that a recalculation of the current time step can be forced using the NUMFLX parameter IRES (see D03PFF or D03PLF).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

GAMMA \leq 0.0

$IFAIL = 2$

On entry,

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

0.0

7 Accuracy

D03PWF performs an exact calculation of the HLL (Harten–Lax–van Leer) numerical flux function, and so the result will be accurate to machine precision.

8 Further Comments

D03PWF 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. The time taken is independent of the input parameters.

9 Example

This example uses D03PLF and D03PWF to solve the Euler equations in the domain

0 \leq x \leq 1

for

0 < t \leq 0.035

with initial conditions for the primitive variables

ρ (x, t)

u (x, t)

and

p (x, t)

given by

\begin{array}{llcrlcrl} ρ (x, 0) = 5.99924, & u (x, 0) = 19.5975, & p (x, 0) = 460.894, & for ​ x < 0.5, \\ ρ (x, 0) = 5.99242, & u (x, 0) = - 6.19633, & p (x, 0) = 46.095, & for ​ x > 0.5 . \end{array}

This test problem is taken from Toro (1996) and its solution represents the collision of two strong shocks travelling in opposite directions, consisting of a left facing shock (travelling slowly to the right), a right travelling contact discontinuity and a right travelling shock wave. There is an exact solution to this problem (see Toro (1996)) but the calculation is lengthy and has therefore been omitted.

NAG Library Routine DocumentD03PWF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D03PWF