NAG CL Interface
d03pwc (dim1_parab_euler_hll)
1
Purpose
d03pwc calculates a numerical flux function using a modified HLL (Harten–Lax–van Leer) Approximate 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: d03pwc, nag_pde_dim1_parab_euler_hll or nag_pde_parab_1d_euler_hll.
3
Description
d03pwc calculates a numerical flux function at a single spatial point using a modified HLL (Harten–Lax–van Leer) Approximate Riemann Solver (see
Toro (1992),
Toro (1996) and
Toro et al. (1994)) 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
d03pwc.
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 an approximation to 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.
4
References
Toro E F (1992) The weighted average flux method applied to the Euler equations Phil. Trans. R. Soc. Lond. A341 499–530
Toro E F (1996) Riemann Solvers and Upwind Methods for Fluid Dynamics Springer–Verlag
Toro E F, Spruce M and Spears W (1994) Restoration of the contact surface in the HLL Riemann solver J. Shock Waves 4 25–34
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 $e$.
Constraints:
 ${\mathbf{uleft}}\left[0\right]\ge 0.0$;
 Left pressure, $\mathit{pl}\ge 0.0$, where $\mathit{pl}$ is calculated using (3).

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 $e$.
Constraints:
 ${\mathbf{uright}}\left[0\right]\ge 0.0$;
 Right pressure, $\mathit{pr}\ge 0.0$, where $\mathit{pr}$ is calculated using (3).

3:
$\mathbf{gamma}$ – double
Input

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

4:
$\mathbf{flux}\left[3\right]$ – double
Output

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

5:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

saved may contain data concerning the computation required by
d03pwc as passed through to
numflx from one of the integrator functions
d03pfc,
d03plc or
d03psc. You should not change the components of
saved.

6:
$\mathbf{fail}$ – 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_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_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, ${\mathbf{uleft}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
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 $\mathit{pr}<0.0$: $\mathit{pr}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
d03pwc performs an exact calculation of the HLL (Harten–Lax–van Leer) numerical flux function, and so the result will be accurate to machine precision.
8
Parallelism and Performance
d03pwc is not threaded in any implementation.
d03pwc 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. The time taken is independent of the input arguments.
10
Example
This example uses
d03plc and
d03pwc 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