NAG Library Routine Document
d03puf (dim1_parab_euler_roe)
1
Purpose
d03puf calculates a numerical flux function using Roe's 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
d03puf calculates a numerical flux function at a single spatial point using Roe's Approximate Riemann Solver (see
Roe (1981)) 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 the routines
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
d03puf.
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 Roe 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. This implementation of Roe's scheme for the Euler equations uses the socalled argumentvector method described in
Roe (1981).
4
References
LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag
Quirk J J (1994) A contribution to the great Riemann solver debate Internat. J. Numer. Methods Fluids 18 555–574
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372
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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 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
d03puf performs an exact calculation of the Roe numerical flux function, and so the result will be accurate to machine precision.
8
Parallelism and Performance
d03puf 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.
d03puf is not threaded in any implementation.
d03puf 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. It should be noted that Roe's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see
Quirk (1994) for examples). In particular Roe's scheme does not satisfy an ‘entropy condition’ which guarantees that the approximate solution of the PDE converges to the correct physical solution, and hence it may admit nonphysical solutions such as expansion shocks. The algorithm used in this routine does not detect or correct any entropy violation. The time taken is independent of the input arguments.
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Example
See
Section 10 in
d03plf.