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 $|v^{n}|v^{n+1}$