NAG Library Routine Document
D03PXF
1 Purpose
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.
2 Specification
INTEGER 
NITER, IFAIL 
REAL (KIND=nag_wp) 
ULEFT(3), URIGHT(3), GAMMA, TOL, FLUX(3) 

3 Description
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:
with
where
$\rho $ is the density,
$m$ is the momentum,
$e$ is the specific total energy and
$\gamma $ is the (constant) ratio of specific heats. The pressure
$p$ is given by
where
$u=m/\rho $ is the velocity.
The routine calculates the numerical flux function
$F\left({U}_{L},{U}_{R}\right)=F\left({U}^{*}\left({U}_{L},{U}_{R}\right)\right)$, where
$U={U}_{L}$ and
$U={U}_{R}$ are the left and right solution values, and
${U}^{*}\left({U}_{L},{U}_{R}\right)$ is the intermediate state
$\omega \left(0\right)$ arising from the similarity solution
$U\left(y,t\right)=\omega \left(y/t\right)$ of the Riemann problem defined by
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.
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 nonvacuum data states. In this case a Riemann solver which can handle vacuum states has to be used (see
Toro (1996)).
4 References
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
5 Parameters
 1: ULEFT($3$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{ULEFT}}\left(\mathit{i}\right)$ must contain the left value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{ULEFT}}\left(1\right)$ must contain the left value of $\rho $, ${\mathbf{ULEFT}}\left(2\right)$ must contain the left value of $m$ and ${\mathbf{ULEFT}}\left(3\right)$ must contain the left value of $e$.
 2: URIGHT($3$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{URIGHT}}\left(\mathit{i}\right)$ must contain the right value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{URIGHT}}\left(1\right)$ must contain the right value of $\rho $, ${\mathbf{URIGHT}}\left(2\right)$ must contain the right value of $m$ and ${\mathbf{URIGHT}}\left(3\right)$ must contain the right value of $e$.
 3: GAMMA – REAL (KIND=nag_wp)Input
On entry: the ratio of specific heats, $\gamma $.
Constraint:
${\mathbf{GAMMA}}>0.0$.
 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
$1.0\times {10}^{6}$ is used.
Constraint:
${\mathbf{TOL}}\ge 0.0$.
 5: NITER – INTEGERInput
On entry: the maximum number of Newton–Raphson iterations allowed. If
NITER is set to zero then the default value of
$20$ is used.
Constraint:
${\mathbf{NITER}}\ge 0$.
 6: FLUX($3$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{FLUX}}\left(\mathit{i}\right)$ contains the numerical flux component ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.
 7: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
Note: if the left and/or right values of
$\rho $ or
$p$ (from
(3)) are found to be negative, then the routine will terminate with an error exit (
${\mathbf{IFAIL}}={\mathbf{2}}$). If the routine is being called from the
NUMFLX etc., then a
soft fail option (
${\mathbf{IFAIL}}={\mathbf{1}}$ or
$1$) 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
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{GAMMA}}\le 0.0$, 
or  ${\mathbf{TOL}}<0.0$, 
or  ${\mathbf{NITER}}<0$. 
 ${\mathbf{IFAIL}}=2$
On entry,  the left and/or right density or derived pressure value is less than $0.0$. 
 ${\mathbf{IFAIL}}=3$
A vacuum condition has been detected therefore a solution cannot be found using this routine. You are advised to check your problem formulation.
 ${\mathbf{IFAIL}}=4$
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).
7 Accuracy
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.
D03PXF must only be used to calculate the numerical flux for the Euler equations in exactly the form given by
(2), with
${\mathbf{ULEFT}}\left(\mathit{i}\right)$ and
${\mathbf{URIGHT}}\left(\mathit{i}\right)$ containing the left and right values of
$\rho ,m$ and
$e$, for
$\mathit{i}=1,2,3$, 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.
9 Example
This example uses
D03PLF and D03PXF to solve the Euler equations in the domain
$0\le x\le 1$ for
$0<t\le 0.035$ with initial conditions for the primitive variables
$\rho \left(x,t\right)$,
$u\left(x,t\right)$ and
$p\left(x,t\right)$ 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.
9.1 Program Text
Program Text (d03pxfe.f90)
9.2 Program Data
Program Data (d03pxfe.d)
9.3 Program Results
Program Results (d03pxfe.r)