Study guide: Intro to Computing with Finite Difference Methods

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The $\theta$ -rule

$\begin{equation} \frac{u^{n+1}-u^n}{\Delta t} = \theta f(u^{n+1},t_{n+1}) + (1-\theta)f(u^n, t_n) \tag{38} \end{equation}$ Bringing the unknown

$u^{n+1}$ to the left-hand side and the known terms on the right-hand side gives

$\begin{equation} u^{n+1} - \Delta t \theta f(u^{n+1},t_{n+1}) = u^n + \Delta t(1-\theta)f(u^n, t_n) \end{equation}$ This is a nonlinear equation in

$u^{n+1}$ (unless

$f$ is linear in

$u$ )!

The θ \theta -rule

The $\theta$ -rule