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2.6.3 Flux of PLM schemes

Improvement over donor cell scheme (constant \bgroup\color{DEFcolor}$\rho$\egroup within cell): piecewise linear method (PLM) - reconstruction by piecewise linear functions.

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} (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)