Von Neumann Analysis

Von Neumann analysis is used to verify the stability of a finite difference scheme (FDS). We will only consider FDSs having 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 FDS 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 FDS for the ideal vibrating string Eq.(N.3):

There is only one spatial dimension, so we only need a single 1D Discrete Time Fourier Transform (DTFT) along [462]. Using the shift theorem for the DTFT, we obtain

where denotes radian spatial frequency (wave number). (On a more elementary level, the DTFT along can be carried out by substituting for in the FDS.) 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 [460], 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 FDS is then declared stable if for all spatial frequencies .

Since the FDS of the ideal vibrating string is so simple, let's find the two poles. Taking the z transform of Eq.(N.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 , we spontaneously have for all . Therefore, we have shown by von Neumann analysis that the FDS Eq.(N.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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.