D03PXF calculates a numerical flux function using an Exact 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.
D03PXF calculates a numerical flux function at a single spatial point using an Exact Riemann Solver (see Toro (1996) and Toro (1989)) 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 D03PXF.
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 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.
The algorithm is termed an Exact Riemann Solver although it does in fact calculate an approximate solution to a true Riemann problem, as opposed to an Approximate Riemann Solver which involves some form of alternative modelling of the Riemann problem. The approximation part of the Exact Riemann Solver is a Newton–Raphson iterative procedure to calculate the pressure, and you must supply a tolerance TOL and a maximum number of iterations NITER. Default values for these parameters can be chosen.
A solution cannot be found by this routine if there is a vacuum state in the Riemann problem (loosely characterised by zero density), or if such a state is generated by the interaction of two non-vacuum data states. In this case a Riemann solver which can handle vacuum states has to be used (see Toro (1996)).
Toro E F (1989) A weighted average flux method for hyperbolic conservation laws Proc. Roy. Soc. Lond.A423 401–418
Toro E F (1996) Riemann Solvers and Upwind Methods for Fluid Dynamics Springer–Verlag
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 .
2: URIGHT() – REAL (KIND=nag_wp) arrayInput
On entry: must contain the right value of the component , for . That is, must contain the right value of , must contain the right value of and must contain the right value of .
3: GAMMA – REAL (KIND=nag_wp)Input
On entry: the ratio of specific heats, .
4: TOL – REAL (KIND=nag_wp)Input
On entry: the tolerance to be used in the Newton–Raphson procedure to calculate the pressure. If TOL is set to zero then the default value of is used.
5: NITER – INTEGERInput
On entry: the maximum number of Newton–Raphson iterations allowed. If NITER is set to zero then the default value of is used.
6: FLUX() – REAL (KIND=nag_wp) arrayOutput
On exit: contains the numerical flux component , for .
7: 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 .
A vacuum condition has been detected therefore a solution cannot be found using this routine. You are advised to check your problem formulation.
The internal Newton–Raphson iterative procedure used to solve for the pressure has failed to converge. The value of TOL or NITER may be too small, but if the problem persists try an Approximate Riemann Solver (D03PUF, D03PVF or D03PWF).
The algorithm is exact apart from the calculation of the pressure which uses a Newton–Raphson iterative procedure, the accuracy of which is controlled by the parameter TOL. In some cases the initial guess for the Newton–Raphson procedure is exact and no further iterations are required.
8 Further Comments
D03PXF 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.
For some problems the routine may fail or be highly inefficient in comparison with an Approximate Riemann Solver (e.g., D03PUF, D03PVF or D03PWF). Hence it is advisable to try more than one Riemann solver and to compare the performance and the results.
The time taken by the routine is independent of all input parameters other than TOL.
This example uses D03PLF and D03PXF 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.