The Euler-Cromer method; idea

2x2 system for $u^{\prime\prime}+\omega^2u=0$: $$\begin{align*} v^{\prime} &= -\omega^2u\\ u^{\prime} &= v \end{align*}$$

Forward-backward discretization:

• Update $v$ with Forward Euler
• Update $u$ with Backward Euler, using latest $v$