Study guide: Solving nonlinear ODE and PDE problems

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Crank-Nicolson discretization

Crank-Nicolson discretization applies a centered difference at $ t_{n+\frac{1}{2}} $: $$ [D_t u = \nabla\cdot (\dfc(u)\nabla u) + f(u)]^{n+\frac{1}{2}}\tp$$

Many choices of formulating an arithmetic means: $$ \begin{align*} [f(u)]^{n+\frac{1}{2}} &\approx f(\frac{1}{2}(u^n + u^{n+1})) = [f(\overline{u}^t)]^{n+\frac{1}{2}}\\ [f(u)]^{n+\frac{1}{2}} &\approx \frac{1}{2}(f(u^n) + f(u^{n+1})) =[\overline{f(u)}^t]^{n+\frac{1}{2}}\\ [\dfc(u)\nabla u]^{n+\frac{1}{2}} &\approx \dfc(\frac{1}{2}(u^n + u^{n+1}))\nabla (\frac{1}{2}(u^n + u^{n+1})) = \dfc(\overline{u}^t)\nabla \overline{u}^t]^{n+\frac{1}{2}}\\ [\dfc(u)\nabla u]^{n+\frac{1}{2}} &\approx \frac{1}{2}(\dfc(u^n) + \dfc(u^{n+1}))\nabla (\frac{1}{2}(u^n + u^{n+1})) = [\overline{\dfc(u)}^t\nabla\overline{u}^t]^{n+\frac{1}{2}}\\ [\dfc(u)\nabla u]^{n+\frac{1}{2}} &\approx \frac{1}{2}(\dfc(u^n)\nabla u^n + \dfc(u^{n+1})\nabla u^{n+1}) = [\overline{\dfc(u)\nabla u}^t]^{n+\frac{1}{2}} \end{align*} $$