NAG CL Interface
d03pxc (dim1_parab_euler_exact)
1
Purpose
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,
d03plc or
d03psc, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
2
Specification
The function may be called by the names: d03pxc, nag_pde_dim1_parab_euler_exact or nag_pde_parab_1d_euler_exact.
3
Description
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,
d03plc and
d03psc, 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:
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 function 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 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 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
Arguments

1:
$\mathbf{uleft}\left[3\right]$ – const double
Input

On entry: ${\mathbf{uleft}}\left[\mathit{i}1\right]$ must contain the left value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uleft}}\left[0\right]$ must contain the left value of $\rho $, ${\mathbf{uleft}}\left[1\right]$ must contain the left value of $m$ and ${\mathbf{uleft}}\left[2\right]$ must contain the left value of $e$.

2:
$\mathbf{uright}\left[3\right]$ – const double
Input

On entry: ${\mathbf{uright}}\left[\mathit{i}1\right]$ must contain the right value of the component ${U}_{\mathit{i}}$, for $\mathit{i}=1,2,3$. That is, ${\mathbf{uright}}\left[0\right]$ must contain the right value of $\rho $, ${\mathbf{uright}}\left[1\right]$ must contain the right value of $m$ and ${\mathbf{uright}}\left[2\right]$ must contain the right value of $e$.

3:
$\mathbf{gamma}$ – double
Input

On entry: the ratio of specific heats, $\gamma $.
Constraint:
${\mathbf{gamma}}>0.0$.

4:
$\mathbf{tol}$ – double
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:
$\mathbf{niter}$ – Integer
Input

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:
$\mathbf{flux}\left[3\right]$ – double
Output

On exit: ${\mathbf{flux}}\left[\mathit{i}1\right]$ contains the numerical flux component ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.

7:
$\mathbf{saved}$ – 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,
d03plc or
d03psc. You should not change the components of
saved.

8:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{niter}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{niter}}\ge 0$.
 NE_INTERNAL_ERROR

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.
 NE_ITER_FAIL_CONV

Newton–Raphson iteration failed to converge.
 NE_NO_LICENCE

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.
 NE_REAL

Left pressure value $\mathit{pl}<0.0$: $\mathit{pl}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{gamma}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{gamma}}>0.0$.
On entry, ${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tol}}\ge 0.0$.
On entry, ${\mathbf{uleft}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{uleft}}\left[0\right]\ge 0.0$.
On entry, ${\mathbf{uright}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{uright}}\left[0\right]\ge 0.0$.
Right pressure value $\mathit{pr}<0.0$: $\mathit{pr}=\u2329\mathit{\text{value}}\u232a$.
 NE_VACUUM

A vacuum condition has been detected.
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 argument
tol. In some cases the initial guess for the Newton–Raphson procedure is exact and no further iterations are required.
8
Parallelism 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
${\mathbf{uleft}}\left[\mathit{i}1\right]$ and
${\mathbf{uright}}\left[\mathit{i}1\right]$ containing the left and right values of
$\rho ,m$ and
$e$, for
$\mathit{i}=1,2,3$, respectively.
For some problems the function may fail or be highly inefficient in comparison with an Approximate Riemann Solver (e.g.,
d03puc,
d03pvc or
d03pwc). 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.
10
Example
This example uses
d03plc and
d03pxc 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.
10.1
Program Text
10.2
Program Data
10.3
Program Results