5.2.3 Crank-Nicholson Scheme (CN)

As before, replace $\partial u/\partial t$ by $\Delta_{n} u/\Delta t \equiv (u^{n+1}-u^{n})/\Delta t$.

Noting that this approximation is in fact centered at $t_{n+1/2}$, introduce the same kind of time centering on the right-hand side: taking the mean of $\delta_{j}^{2}u^{n}$ (= FTCS) and $\delta_{j}^{2}u^{n+1}$ (= implicit scheme) we write

$$\frac{1}{\Delta t} \left[ u_{j}^{n+1}-u_{j}^{n}\right] = \frac{\lambda}{2(\Delta x)^{2}} \left[ (u_{j+1}^{n+1}-2 u_{j}^{n+1}+u_{j-1}^{n+1}) + (u_{j+1}^{n}-2 u_{j}^{n}+u_{j-1}^{n}) \right]$$

The Crank-Nicholson formula is of second order in $\Delta t$.

Defining $a \equiv \lambda \Delta t / 2(\Delta x)^{2}$ we may write CN as

$$\fbox{$ \displaystyle -a u_{j-1}^{n+1}+(1+2a)u_{j}^{n+1}-a u_{j+1}^{n+1}= a u_{j-1}^{n}+(1-2a)u_{j}^{n}+a u_{j+1}^{n} $}$$

or

$$\mbox{${\bf A}$} \cdot \mbox{$\bf u$}^{n+1} = \mbox{${\bf B}$} \cdot \mbox{$\bf u$}^{n}$$

with

$$\mbox{${\bf A}$} \!\equiv \! \left( \begin{array}{cccccc} ... ...& 1\!-\!2a & a \\ . & . & . & 0 & 0 & 1 \end{array}\right)$$

Tridiagonal $\Longrightarrow$Solve by Recursion!

Stability of CN:

The amplification factor is

$$g(k) = \frac{1-2a\,\sin^{2}(k\Delta x/2)}{1+2a\,\sin^{2}(k\Delta x/2)} \leq 1 \; ,$$

which makes the CN method unconditionally stable.



EXAMPLE: The time-dependent Schroedinger equation,

$$\frac{\partial u}{\partial t} = -i H u \; , \;\;\; {\rm with} \;\; H \equiv \frac{\partial^{2}}{\partial x^{2}} + U(x)$$

when rewritten à la Crank-Nicholson, reads

$$\frac{1}{\Delta t}[u_{j}^{n+1}-u_{j}^{n}] = - \frac{i}{2} [(Hu)_{j}^{n+1}+(Hu)_{j}^{n}]$$

$$= - \frac{i}{2} \left[ \frac{\delta_{j}^{2}u_{j}^{n+1}}{(\Delta x)^... ...}^{n+1} +\frac{\delta_{j}^{2}u_{j}^{n}}{(\Delta x)^{2}} +U_{j}u_{j}^{n} \right]$$

With $a \equiv \Delta t/2 (\Delta x)^{2}$ and $b_{j} \equiv U(x_{j}) \Delta t/2$ this leads to

$$(ia) u_{j-1}^{n+1}+(1-2ia+ib_{j})u_{j}^{n+1} + (ia) u_{j+1}^{n+1}=$$

$$=(-ia) u_{j-1}^{n}+(1+2ia-ib_{j})u_{j}^{n}+(-ia) u_{j+1}^{n}$$

Again, we have a tridiagonal system which may be inverted very efficiently by recursion.