$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Staggered grid
- \( u \) is unknown at \( t_n \): \( u^n \)
- \( v \) is unknown at \( t_{n+1/2} \): \( v^{n+\half} \)
- All derivatives are approximated by centered differences
$$
\begin{align*}
\lbrack D_t u &= v\rbrack^{n-\half}
\\
\lbrack D_tv &= m^{-1}\left(F(t) - f(v) - s(u)\right)\rbrack^n
\end{align*}
$$
Written out,
$$
\begin{align*}
\frac{u^n - u^{n-1}}{\Delta t} &= v^{n-\half}\\
\frac{v^{n+\half} - v^{n-\half}}{\Delta t}
&= m^{-1}\left(F^n - f(v^n) - s(u^n)\right)
\end{align*}
$$
Problem: \( f(v^n) \)
