2.6.3 Examples: PLM

Once we know the cell average $\bgroup\color{DEFcolor}$\rho$\egroup$ and the slope $\bgroup\color{DEFcolor}$\delta \! \rho$\egroup$ , we get the flux over $\bgroup\color{DEFcolor}$\Delta t$\egroup$ from

$\begin{displaymath} f_{i \pm \frac{1}{2}} = v_{} \left\{ \rho_i + \frac{... ...ight) \frac{\Delta t}{\Delta x} \right] \right\} \enspace . \end{displaymath}$

(134)

Minmod (minimum allowed slope):

$\displaystyle \delta \! \rho_i$	$\textstyle =$	$\displaystyle \min \! \left( \max \! \left( \rho_{i} - \rho_{i-1}, 0 \right), \max \! \left( \rho_{i+1} - \rho_{i} , 0 \right) \right) +$	(135)
		$\displaystyle \max \! \left( \min \! \left( \rho_{i} - \rho_{i-1}, 0 \right), \min \! \left( \rho_{i+1} - \rho_{i} , 0 \right) \right)$

vanLeer (harmonic mean of slopes):

$\begin{displaymath} \delta \! \rho_i = \left\{ \begin{array}{ll} \frac{1}{\fr... ...i} \right) > 0 \\ 0 & \mbox{elsewhere} \end{array} \right. \end{displaymath}$

(136)

Superbee (maximum allowed slope):

$\displaystyle \delta \! \rho_i$	$\textstyle =$	$\displaystyle \left[ \mbox{sign} \! \left( \rho_{i} - \rho_{i-1} \right) + \mbox{sign} \! \left( \rho_{i+1} - \rho_{i} \right) \right]$	(137)
		$\displaystyle \min \! \left( \mbox{abs} \! \left( \rho_{i} - \rho_{i-1} \right)... ...1} \right) , \mbox{abs} \! \left( \rho_{i+1} - \rho_{i} \right) \right) \right)$

piecewise linear reconstruction

slope-limiter

(flux-limiter)