Loading [MathJax]/extensions/TeX/boldsymbol.js
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
A centered scheme on a staggered mesh
Staggered mesh:
- u is computed at mesh points t_n
- v is computed at points t_{n+\half}
Centered differences in (42)-(42):
\begin{align}
[D_t u &= v]^{n-\half},
\tag{48} \\
[D_t v &= \frac{1}{m}( F(t) - \beta |v|v - s(u))]^{n}\tp
\tag{49}
\end{align}
- Problem: |v^n|v^n , because v^n is not computed directly
- Remedy: Geometric mean,
|v^n|v^n \approx |v^{n-\half}|v^{n+\half}\tp
