# NAG CL Interfaced03pvc (dim1_​parab_​euler_​osher)

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## 1Purpose

d03pvc calculates a numerical flux function using Osher's 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.

## 2Specification

 #include
 void d03pvc (const double uleft[], const double uright[], double gamma, Nag_OsherVersion path, double flux[], Nag_D03_Save *saved, NagError *fail)
The function may be called by the names: d03pvc, nag_pde_dim1_parab_euler_osher or nag_pde_parab_1d_euler_osher.

## 3Description

d03pvc calculates a numerical flux function at a single spatial point using Osher's Approximate Riemann Solver (see Hemker and Spekreijse (1986) and Pennington and Berzins (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 the functions 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 d03pvc.
The Euler equations for a perfect gas in conservative form are:
 $∂U ∂t + ∂F ∂x =0,$ (1)
with
 (2)
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
 $p=(γ-1) (e-ρu22) ,$ (3)
where $u=m/\rho$ is the velocity.
The function calculates the Osher 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
 $∂U ∂t + ∂F ∂y =0,$ (4)
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 , where $y=0$ is the point at which the numerical flux is required. Osher's solver carries out an integration along a path in the phase space of $U$ consisting of subpaths which are piecewise parallel to the eigenvectors of the Jacobian of the PDE system. There are two variants of the Osher solver termed O (original) and P (physical), which differ in the order in which the subpaths are taken. The P-variant is generally more efficient, but in some rare cases may fail (see Hemker and Spekreijse (1986) for details). The argument path specifies which variant is to be used. The algorithm for Osher's solver for the Euler equations is given in detail in the Appendix of Pennington and Berzins (1994).

## 4References

Hemker P W and Spekreijse S P (1986) Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations Applied Numerical Mathematics 2 475–493
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99
Quirk J J (1994) A contribution to the great Riemann solver debate Internat. J. Numer. Methods Fluids 18 555–574

## 5Arguments

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: $\mathbf{path}$Nag_OsherVersion Input
On entry: the variant of the Osher scheme.
${\mathbf{path}}=\mathrm{Nag_OsherOriginal}$
Original.
${\mathbf{path}}=\mathrm{Nag_OsherPhysical}$
Physical.
Constraint: ${\mathbf{path}}=\mathrm{Nag_OsherOriginal}$ or $\mathrm{Nag_OsherPhysical}$.
5: $\mathbf{flux}\left[3\right]$double Output
On exit: ${\mathbf{flux}}\left[\mathit{i}-1\right]$ contains the numerical flux component ${\stackrel{^}{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,3$.
6: $\mathbf{saved}$Nag_D03_Save * Communication Structure
saved may contain data concerning the computation required by d03pvc as passed through to numflx from one of the integrator functions d03pfc, d03plc or d03psc. You should not change the components of saved.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ 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 $\mathit{pl}<0.0$: $\mathit{pl}=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{gamma}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{gamma}}>0.0$.
On entry, ${\mathbf{uleft}}\left[0\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{uleft}}\left[0\right]\ge 0.0$.
On entry, ${\mathbf{uright}}\left[0\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{uright}}\left[0\right]\ge 0.0$.
Right pressure value $\mathit{pr}<0.0$: $\mathit{pr}=⟨\mathit{\text{value}}⟩$.

## 7Accuracy

d03pvc performs an exact calculation of the Osher numerical flux function, and so the result will be accurate to machine precision.

## 8Parallelism and Performance

d03pvc 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. It should be noted that Osher's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see Quirk (1994) for examples). The time taken depends on the input argument path and on the left and right solution values, since inclusion of each subpath depends on the signs of the eigenvalues. In general this cannot be determined in advance.