naginterfaces.library.pde.dim1_​parab_​euler_​hll

naginterfaces.library.pde.dim1_parab_euler_hll(uleft, uright, gamma, comm)

dim1_parab_euler_hll 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 dim1_parab_convdiff(), dim1_parab_convdiff_dae() or dim1_parab_convdiff_remesh(), but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

For full information please refer to the NAG Library document for d03pw

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d03/d03pwf.html

Parameters

uleftfloat, array-like, shape $\left(3\right)$

$uleft[\text{i}-1]$ must contain the left value of the component $U_{\text{i}}$, for $\text{i}=1,2,\dots ,3$. That is, $uleft\left[0\right]$ must contain the left value of $\rho$, $uleft\left[1\right]$ must contain the left value of $m$ and $uleft\left[2\right]$ must contain the left value of $e$.

urightfloat, array-like, shape $\left(3\right)$

$uright[\text{i}-1]$ must contain the right value of the component $U_{\text{i}}$, for $\text{i}=1,2,\dots ,3$. That is, $uright\left[0\right]$ must contain the right value of $\rho$, $uright\left[1\right]$ must contain the right value of $m$ and $uright\left[2\right]$ must contain the right value of $e$.

gammafloat

The ratio of specific heats, $\gamma$.

commdict, communication object

Communication structure.

On initial entry: need not be set.

Returns

fluxfloat, ndarray, shape $\left(3\right)$

$flux[\text{i}-1]$ contains the numerical flux component ${\hat{F}}_{\text{i}}$, for $\text{i}=1,2,\dots ,3$.

Raises

NagValueError

(errno $1$)

On entry, $gamma=⟨value⟩$.

Constraint: $gamma>0.0$.

(errno $2$)

Right pressure value $\text{pr}<0.0$: $\text{pr}=⟨value⟩$.

(errno $2$)

Left pressure value $\text{pl}<0.0$: $\text{pl}=⟨value⟩$.

(errno $2$)

On entry, $uright\left[0\right]=⟨value⟩$.

Constraint: $uright\left[0\right]\ge 0.0$.

(errno $2$)

On entry, $uleft\left[0\right]=⟨value⟩$.

Constraint: $uleft\left[0\right]\ge 0.0$.

Notes

dim1_parab_euler_hll 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 dim1_parab_convdiff(), dim1_parab_convdiff_dae() and dim1_parab_convdiff_remesh(), the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the function argument $\text{numflx}$ from which you may call dim1_parab_euler_hll.

The Euler equations for a perfect gas in conservative form are:

$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}=0\text{,}$$

with

$$\begin{array}{c}U=\left[\begin{array}{c}\rho \\ m\\ e\end{array}\right]\text{and}F=\left[\begin{array}{c}m\\ \frac{m^2}{\rho }+(\gamma -1)(e-\frac{m^2}{2\rho })\\ \frac{me}{\rho }+\frac{m}{\rho }(\gamma -1)(e-\frac{m^2}{2\rho })\end{array}\right]\text{,}\end{array}$$

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

$$p=(\gamma -1)(e-\frac{\rho u^2}{2})\text{,}$$

where $u=m/\rho$ is the velocity.

The function calculates an approximation to the numerical flux function $F(U_L,U_R)=F\left(U^{\ast }(U_L,U_R)\right)$, where $U=U_L$ and $U=U_R$ are the left and right solution values, and $U^{\ast }(U_L,U_R)$ is the intermediate state $\omega \left(0\right)$ arising from the similarity solution $U(y,t)=\omega \left(y/t\right)$ of the Riemann problem defined by

$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial y}=0\text{,}$$

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.

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