4.3 A standard implementation of a HRSC scheme

$$\begin{eqnarray*} \frac{\partial{\bf u}}{\partial t} + \frac{\partial{\bf f}({\bf u})}{\partial x} = 0 \hspace{1cm} {\bf f}={\bf f}({\bf u}) \end{eqnarray*}$$

1. Numerical grid:

2. Time update: $\rightarrow$ Conservation form algorithm

$$\begin{eqnarray*} {\bf u}_j^{n+1} = {\bf u}_j^n - \frac{\Delta t}{\Delta x} \left(\hat{\bf f}_{j+\frac{1}{2}} - \hat{\bf f}_{j-\frac{1}{2}}\right) \end{eqnarray*}$$

In practice: 2nd or 3rd order time accurate conservative Runge-Kutta schemes (Shu and Osher 1989)

$t^n \rightarrow t^* \rightarrow t^{**} \rightarrow \cdots \rightarrow t^{n+1}$

$$\begin{eqnarray*} L\equiv -\frac{\hat{\bf f}_{j+{1\over 2}}- \hat{\bf f}_{j-{1\over 2}}} {\Delta x} \end{eqnarray*}$$

$$\begin{eqnarray*} {\bf u}^{(1)}={\bf u}^{n} + \Delta t L({\bf u}^{n}) \end{eqnarray*}$$

$$\begin{eqnarray*} {\bf u}^{(2)}={\bf u}^{n} + \frac{1}{4} \Delta t L({\bf u}^{n}) + \frac{1}{4} \Delta t L({\bf u}^{(1)}) \end{eqnarray*}$$

$$\begin{eqnarray*} {\bf u}^{n+1}={\bf u}^{n} + \frac{1}{6} \Delta t L({\bf u}^{n... ...lta t L({\bf u}^{(1)}) + \frac{2}{3} \Delta t L({\bf u}^{(2)}) \end{eqnarray*}$$

3. Numerical fluxes:

Approximate Riemann solvers: Roe, HLLE, Marquina

Explicit use of the characteristic information of the system

Flux Formula

$$\begin{eqnarray*} \widehat{{\bf f}}_{j\pm{1\over 2}} = \frac{1}{2} &\left( {\b... ...parrow& \\ &\hspace{3.7cm}\textrm{\bf \lq\lq numerical viscosity''}& \end{eqnarray*}$$

${\bf u}^L$, ${\bf u}^R$: left and right ``reconstructed" variables
$\lambda$: eigenvalues
$\Delta {\omega}$: jump of characteristic variables
$r$: right-eigenvectors

4. Cell reconstruction:

Piecewise constant (Godunov) or linear (``MUSCL" scheme, van Leer) interpolation procedures of ${\bf u}$ from cell centers to cell interfaces. Higher-order procedures also available: parabolic (PPM, Colella and Woodward) or hyperbolic (PHM, Marquina)

$$\begin{eqnarray*} {\bf u}^{L,R}_j = {\bf u}_j + s_j(x_{j\pm\frac{1}{2}}-x_j) \hspace{2cm} x_j=\frac{1}{2}(x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}) \end{eqnarray*}$$

with

$$\begin{eqnarray*} s_j= \left\{ \begin{array}{cc} \min(\Delta Q^{j+\frac{1}{2}},... ...^{j-\frac{1}{2}})$} \\ 0 & \mbox{otherwise} \end{array}\right. \end{eqnarray*}$$

with

$$\begin{eqnarray*} \Delta Q^{j+\frac{1}{2}}=\frac{{\bf u}_{j+1}-{\bf u}_j}{x_{j+1}-x_{j}} \end{eqnarray*}$$