## 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):

*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

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

*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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