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##

5.2.3 Crank-Nicholson Scheme (CN)

As before, replace
by
.

Noting that this approximation is in fact centered at ,
introduce the same kind of time centering on the right-hand side:
taking the
mean of
(= FTCS) and
(= implicit scheme) we write

The *Crank-Nicholson* formula is of
*second* order in .

Defining
we may write CN as

or

with

*Tridiagonal*
*Solve by Recursion!*

*Stability of CN:*

The amplification factor is

which makes the CN method unconditionally stable.

__EXAMPLE:__
The time-dependent Schroedinger equation,

when rewritten à la Crank-Nicholson, reads

With
and
this leads to

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

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**Up:** 5.2 Initial Value Problems
** Previous:** 5.2.2 Implicit Scheme of
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001