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:

../_images/Implicit_Stability_Diagram.png

Fig. 6 The amplification factor of the general implicit scheme, for a range of CFL numbers and \(\alpha\).#

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.

../_images/ImplicitScheme_Solution.png

Fig. 7 The solution of the implicit scheme after 10 time steps for a top hat initial condition, for CFL numbers of 1.9 and \(\alpha=0.5, 0.7, 0.9, 0.99\), with \(\Delta t=2\) and \(\Delta x=1\).#