$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
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,
$$ |v^n|v^n \approx |v^{n-\half}|v^{n+\half}\tp$$
