NAG Library Routine Document
d03pwf (dim1_parab_euler_hll)
1
Purpose
d03pwf 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
d03pff,
d03plf or
d03psf, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
2
Specification
Fortran Interface
Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  uleft(3), uright(3), gamma  Real (Kind=nag_wp), Intent (Out)  ::  flux(3) 

3
Description
d03pwf 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
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
d03pwf.
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 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: $\mathbf{uleft}\left(3\right)$ – 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 $e$.
Constraints:
 ${\mathbf{uleft}}\left(1\right)\ge 0.0$;
 Left pressure, $\mathit{pl}\ge 0.0$, where $\mathit{pl}$ is calculated using (3).
 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 $e$.
Constraints:
 ${\mathbf{uright}}\left(1\right)\ge 0.0$;
 Right pressure, $\mathit{pr}\ge 0.0$, where $\mathit{pr}$ is calculated using (3).
 3: $\mathbf{gamma}$ – Real (Kind=nag_wp)Input

On entry: the ratio of specific heats, $\gamma $.
Constraint:
${\mathbf{gamma}}>0.0$.
 4: $\mathbf{flux}\left(3\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: ${\mathbf{flux}}\left(\mathit{i}\right)$ contains the numerical flux component ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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 argument
ires (see
d03pff or
d03plf).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, 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,  ${\mathbf{gamma}}\le 0.0$. 
 ${\mathbf{ifail}}=2$

On entry,  the left and/or right density or derived pressure value is less than $0.0$. 
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
d03pwf 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
d03pwf is not thread safe and should not be called from a multithreaded user program. Please see
Section 3.12.1 in How to Use the NAG Library and its Documentation for more information on thread safety.
d03pwf is not threaded in any implementation.
d03pwf 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. The time taken is independent of the input arguments.
10
Example
This example uses
d03plf and
d03pwf 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
Program Text (d03pwfe.f90)
10.2
Program Data
Program Data (d03pwfe.d)
10.3
Program Results
Program Results (d03pwfe.r)