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,d03plford03psf, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
The routine may be called by the names d03pxf or nagf_pde_dim1_parab_euler_exact.
3Description
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,d03plfandd03psf, 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 $\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
The routine calculates the numerical flux function $F({U}_{L},{U}_{R})=F\left({U}^{*}({U}_{L},{U}_{R})\right)$, 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 $\omega \left(0\right)$ arising from the similarity solution $U(y,t)=\omega (y/t)$ 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 arguments 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)).
4References
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
5Arguments
1: $\mathbf{uleft}\left(3\right)$ – 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: $\mathbf{uright}\left(3\right)$ – 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: $\mathbf{gamma}$ – Real (Kind=nag_wp)Input
On entry: the ratio of specific heats, $\gamma $.
Constraint:
${\mathbf{gamma}}>0.0$.
4: $\mathbf{tol}$ – 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 $1.0\times {10}^{\mathrm{-6}}$ is used.
Constraint:
${\mathbf{tol}}\ge 0.0$.
5: $\mathbf{niter}$ – 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: $\mathbf{flux}\left(3\right)$ – 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: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 $\mathrm{-1}$) is recommended so that a recalculation of the current time step can be forced using the numflx argument ires (see d03pfford03plf).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{gamma}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{gamma}}>0.0$.
On entry, ${\mathbf{niter}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{niter}}\ge 0$.
On entry, ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tol}}\ge 0.0$.
${\mathbf{ifail}}=2$
Left pressure value $\mathit{pl}<0.0$: $\mathit{pl}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{uleft}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{uleft}}\left(1\right)\ge 0.0$.
On entry, ${\mathbf{uright}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{uright}}\left(1\right)\ge 0.0$.
Right pressure value $\mathit{pr}<0.0$: $\mathit{pr}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
A vacuum condition has been detected.
${\mathbf{ifail}}=4$
Newton–Raphson iteration failed to converge.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
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
d03pxf is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.
d03pxf is not threaded in any implementation.
9Further Comments
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,d03pvford03pwf). 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 arguments other than tol.
10Example
This example uses d03plfandd03pxf 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 $\rho (x,t)$, $u(x,t)$ and $p(x,t)$ 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.