$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The forward-backward scheme
Forward step for \( u \), backward step for \( v \):
$$
\begin{align}
[D_t^+ u &= v]^n,
\tag{44} \\
[D_t^-v &= \frac{1}{m}( F(t) - \beta |v|v - s(u))]^{n+1}\tp
\tag{45}
\end{align}
$$
- Note:
- step \( u \) forward with known \( v \) in (44)
- step \( v \) forward with known \( u \) in (45)
- Problem: \( |v|v \) gives nonlinearity \( |v^{n+1}|v^{n+1} \).
- Remedy: linearized as \( |v^{n}|v^{n+1} \)
$$
\begin{align}
[D_t^+ u &= v]^n,
\tag{46} \\
[D_t^-v]^{n+1} &= \frac{1}{m}( F(t_{n+1}) - \beta |v^n|v^{n+1} - s(u^{n+1}))\tp
\tag{47}
\end{align}
$$
