Von Neumann Analysis
Von Neumann analysis is used to verify the
stability
of a
finite-difference scheme. We will only consider
one time dimension, but any number of spatial dimensions.

The procedure, in principle, is to perform a
spatial Fourier
transform along all spatial dimensions, thereby reducing the
finite-difference scheme to a time recursion in terms of the spatial
Fourier transform of the system. The system is then stable if this
time recursion is at least marginally stable as a
digital filter.
Let's apply von Neumann analysis to the finite-difference scheme for
the
ideal vibrating string Eq.

(
D.3):
There is only one spatial dimension, so we only need a single 1D
Discrete Time Fourier Transform (
DTFT) along

[
451].
Using the
shift theorem for the DTFT, we obtain
where

denotes radian
spatial frequency (wavenumber).
(On a more elementary level, the DTFT along

can be carried out by
substituting

for

in the finite-difference scheme.)
This is now
a second-order
difference equation (digital
filter) that needs its
stability checked. This can be accomplished most easily using the
Durbin recursion [
449], or we can check that the
poles of the
recursion do not lie outside the unit circle in the

plane.
A method equivalent to checking the pole radii, and typically used
when the time recursion is first order, is to compute the
amplification factor as the complex gain

in
the relation
The finite-difference scheme is then declared stable if

for all spatial frequencies

.
Since the finite-difference scheme of the ideal
vibrating string is so
simple, let's find the two poles. Taking the
z transform of Eq.

(
D.8)
yields
yielding the following characteristic polynomial:
Applying the
quadratic formula to find the roots yields
The squared pole moduli are then given by
Thus, for marginal stability, we require

, and the
poles become
Since the range of spatial frequencies is
![$ k\in[-\pi/X,\pi/X]$](http://www.dsprelated.com/josimages_new/pasp/img4504.png)
, we
spontaneously have

for all

. Therefore, we have shown
by von Neumann analysis that the finite-difference scheme Eq.

(
D.3)
for the ideal vibrating string is
stable.
In summary, von Neumann analysis verifies that no spatial Fourier
components in the system are growing exponentially with respect to
time.
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