2.3.10 Stencil diagrams

**Figure 6:** Stencil diagram.
$\textstyle \parbox{3.0cm}{\makebox[3.0cm]{}}$ $\includegraphics[width=7.5cm]{images/pde1d_stencil_mm0pp.ps}$ $\textstyle \parbox{3.0cm}{\makebox[3.0cm]{}}$

The density $\bgroup\color{DEFcolor}$\rho_i^{n+1}$\egroup$ at grid point $\bgroup\color{DEFcolor}$i$\egroup$ and time-step $\bgroup\color{DEFcolor}$n+1$\egroup$ depends on values at the old time-step $\bgroup\color{DEFcolor}$n$\egroup$ (direct numerical domain of dependence, stencil)

$\begin{displaymath} \rho_{i-k}^n, \rho_{i-k+1}^n, \ldots, \rho_{i+l}^n \enspace . \end{displaymath}$

(103)

This is sketched in a so-called stencil diagram as in Fig. 6.

Looking the other way, a stencil diagram tells also which points

$\begin{displaymath} \rho_{i-l}^{n+1}, \rho_{i-l+1}^{n+1}, \ldots, \rho_{i+k}^{n+1} \end{displaymath}$

(104)

at the new time-step $\bgroup\color{DEFcolor}$n+1$\egroup$ are influenced by $\bgroup\color{DEFcolor}$\rho_i^n$\egroup$ (range of influence).