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.
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:
where is the density, is the momentum, is the specific total energy and is the (constant) ratio of specific heats. The pressure is given by
where is the velocity.
The routine calculates an approximation to the numerical flux function , where and are the left and right solution values, and is the intermediate state arising from the similarity solution of the Riemann problem defined by
with and as in (2), and initial piecewise constant values for and for . The spatial domain is , where is the point at which the numerical flux is required.
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 Waves4 25–34
1: ULEFT() – REAL (KIND=nag_wp) arrayInput
On entry: must contain the left value of the component , for . That is, must contain the left value of , must contain the left value of and must contain the left value of .
On exit: contains the numerical flux component , for .
5: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . 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 is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Note: if the left and/or right values of or (from (3)) are found to be negative, then the routine will terminate with an error exit (). If the routine is being called from the NUMFLX etc., then a soft fail option ( or ) 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 or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
the left and/or right density or derived pressure value is less than .
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 and containing the left and right values of and , for , respectively. The time taken is independent of the input parameters.
This example uses D03PLF and D03PWF to solve the Euler equations in the domain for with initial conditions for the primitive variables , and given by
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.