The Implicit Scheme
Contents
The Implicit Scheme#
We can derive a general implict scheme by noting that the Crank-Nicholson scheme has the spatial derivatives at \(t^n\) and \(t^{n+1}\) multiplied by \(1/2\). We can generalize the formula in the following fashion:
(56)#\[
\frac{u_m^{n+1}-u_m^n}{\Delta t} = -\frac{c}{2}\left(\alpha\frac{u_{m+1}^{n+1}-u_{m-1}^{n+1}}{2\Delta x} + (1-\alpha)\frac{u_{m+1}^{n}-u_{m-1}^{n}}{2\Delta x}\right)
\]
The value of \(\alpha\) determines the nature of the scheme:
(57)#\[\begin{align}
\alpha &= 0 \quad \text{Fully explicit scheme} \\
\alpha &= 0.5 \quad \text{Crank-Nicholson scheme} \\
\alpha &= 1 \quad \text{Fully implicit scheme}
\end{align}\]
Consistency, stability and convergence#
The formula of the amplification factor (55) can be generalized following (56):
(58)#\[
G = \frac{u_{m}^{n+1}}{u_{m}^{n}} = \frac{1-(1-\alpha)(\sigma/2)i\sin\lambda \Delta x}{1+\alpha(\sigma/2)i\sin\lambda \Delta x},
\]
The effect of varying \(\alpha\) in the stability is shown below:
The implicit scheme \(\alpha > 0.5\) is damping and for \(\alpha < 0.5\), the scheme is unconditionally unstable. The damping effect of the numerical diffusion of the implicit scheme increases as the CFL number increases.