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,d03plford03psf, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
The routine may be called by the names d03pwf or nagf_pde_dim1_parab_euler_hll.
3Description
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,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 d03pwf.
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 an approximation to 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.
4References
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 Waves4 25–34
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 $e$.
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 $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ 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 $-1$) 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 $-1$, 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: ${\mathbf{gamma}}>0.0$.
${\mathbf{ifail}}=2$
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 $\mathit{pr}<0.0$: $\mathit{pr}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=-99$
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
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.
8Parallelism and Performance
d03pwf 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.
d03pwf is not threaded in any implementation.
9Further Comments
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.
10Example
This example uses d03plfandd03pwf 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.