d03pxc 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 d03pfc,d03plcord03psc, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
The function may be called by the names: d03pxc, nag_pde_dim1_parab_euler_exact or nag_pde_parab_1d_euler_exact.
d03pxc 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 d03pfc,d03plcandd03psc, 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 d03pxc.
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 function 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 arguments can be chosen.
A solution cannot be found by this function 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: – const doubleInput
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: – const doubleInput
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: – doubleInput
On entry: the ratio of specific heats, .
4: – doubleInput
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: – 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: – doubleOutput
On exit: contains the numerical flux component , for .
7: – Nag_D03_Save *Communication Structure
saved may contain data concerning the computation required by d03pxc as passed through to numflx from one of the integrator functions d03pfc,d03plcord03psc. You should not change the components of saved.
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
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.
Newton–Raphson iteration failed to converge.
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.
Left pressure value : .
On entry, .
On entry, .
On entry, .
On entry, .
Right pressure value : .
A vacuum condition has been detected.
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 argument tol. In some cases the initial guess for the Newton–Raphson procedure is exact and no further iterations are required.
8Parallelism and Performance
d03pxc is not threaded in any implementation.
d03pxc 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 function may fail or be highly inefficient in comparison with an Approximate Riemann Solver (e.g., d03puc,d03pvcord03pwc). Hence it is advisable to try more than one Riemann solver and to compare the performance and the results.
The time taken by the function is independent of all input arguments other than tol.
This example uses d03plcandd03pxc 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.