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2.3.10 Stencil diagrams

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

\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).