2.6.3 Flux of PLM schemes

Donor cell scheme: constant $\rho$ within each cell
Improvement: piecewise linear method (PLM): reconstruction by linear function in each cell

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

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

(147)

If the boundary value in the cell is used for the entire cell we get

$\begin{displaymath} f_{i \pm \frac{1}{2}} = v_{} \left\{ \rho_i + \frac{... ..._i \mbox{sign} \! \left( v_{} \right) \right\} \enspace . \end{displaymath}$

(148)

For $\bgroup\color{DEFcolor}$v_{} > 0$\egroup$ we get from each cell $\bgroup\color{DEFcolor}$\rho_i$\egroup$ the flux $\bgroup\color{DEFcolor}$f_{i + \frac{1}{2}}$\egroup$ .

For $\bgroup\color{DEFcolor}$ v_{} < 0$\egroup$ we get from each cell $\bgroup\color{DEFcolor}$\rho_i$\egroup$ the flux $\bgroup\color{DEFcolor}$f_{i - \frac{1}{2}}$\egroup$ .