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The unifying \theta -rule
The Forward Euler, Backward Euler, and Crank-Nicolson schemes can be
formulated as one scheme with a varying parameter \theta :
\begin{equation}
\frac{u^{n+1}-u^{n}}{t_{n+1}-t_n} = -a (\theta u^{n+1} + (1-\theta) u^{n})
\tag{13}
\end{equation}
- \theta =0 : Forward Euler
- \theta =1 : Backward Euler
- \theta =1/2 : Crank-Nicolson
- We may alternatively choose any \theta\in [0,1] .
u^n is known, solve for u^{n+1} :
\begin{equation}
u^{n+1} = \frac{1 - (1-\theta) a(t_{n+1}-t_n)}{1 + \theta a(t_{n+1}-t_n)}
\tag{14}
\end{equation}
