Study guide: Solving nonlinear ODE and PDE problems

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Example: Crank-Nicolson scheme for the oscillating pendulum model

The scaled equations for an oscillating pendulum: $$ \begin{align} \dot\omega &= -\sin\theta -\beta \omega |\omega|,\\ \dot\theta &= omega, \end{align} $$

Set $ u_0=\omega $, $ u_1=\theta $ $$ \begin{align*} u_0^{\prime} = f_0(u,t) &= -\sin u_1 - \beta u_0|u_0|,\\ u_1^{\prime} = f_1(u,t) &= u_1\tp \end{align*} $$

Crank-Nicolson discretization: $$ \begin{align} \frac{u_0^{n+1}-u_0^{n}}{\Delta t} &= -\sin u_1^{n+\frac{1}{2}} - \beta u_0^{n+\frac{1}{2}}|u_0^{n+\frac{1}{2}}| \approx -\sin\left(\frac{1}{2}(u_1^{n+1} + u_1n)\right) - \beta\frac{1}{4} (u_0^{n+1} + u_0^n)|u_0^{n+1}+u_0^n|,\\ \frac{u_1^{n+1}-u_1^n}{\Delta t} &= v_0^{n+\frac{1}{2}}\approx \frac{1}{2} (u_0^{n+1}+u_0^n)\tp \end{align} $$