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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. (L.3):

$\displaystyle y_{n+1,m}= y_{n,m+1}+ y_{n,m-1}- y_{n-1,m} \protect$

There is only one spatial dimension, so we only need a single 1D Discrete Time Fourier Transform (DTFT) along $ m$ [422]. Using the shift theorem for the DTFT, we obtain
$\displaystyle Y_{n+1}(k)$ $\displaystyle =$ $\displaystyle (e^{jkX} + e^{-jkX})Y_n(k) - Y_{n-1}(k)$  
  $\displaystyle =$ $\displaystyle 2\cos(kX)Y_n(k) - Y_{n-1}(k)$  
  $\displaystyle \isdef$ $\displaystyle 2c_kY_n(k) - Y_{n-1}(k)
\protect$ (L.8)

where $ k=2\pi/\lambda$ denotes radian spatial frequency (wave number). (On a more elementary level, the DTFT along $ m$ can be carried out by substituting $ Y_n(k)e^{jkX}$ for $ y(n,m)$ 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 [426], or we can check that the poles of the recursion do not lie outside the unit circle in the $ z$ 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 $ G(k)$ in the relation

$\displaystyle Y_{n+1}(k) = G(k)Y_n(k).
$

The FDS is then declared stable if $ \vert G(k)\vert\leq 1$ for all spatial frequencies $ k$.

Since the FDS of the ideal vibrating string is so simple, let's find the two poles. Taking the z transform of Eq. (L.8) yields

$\displaystyle zY(z,k) = 2c_k Y(z,k) - z^{-1}Y(z,k)
$

yielding the following characteristic polynomial:

$\displaystyle z^2 - 2c_k z - 1 = 0
$

Applying the quadratic formula to find the roots yields

$\displaystyle z = c_k \pm \sqrt{c_k^2 - 1}.
$

The squared pole moduli are then given by

$\displaystyle \left\vert z\right\vert^2 = c_k^2 \pm (c_k^2 - 1) =
\left\{\begi...
...eq 1 \\ [5pt]
[1,1], & \left\vert c_k\right\vert\leq 1 \\
\end{array}\right..
$

Thus, for marginal stability, we require $ \left\vert c_k\right\vert\leq 1$, and the poles become

$\displaystyle z = c_k \pm j\sqrt{1-c_k^2} = \cos(kX) \pm j\sin(kX) = e^{\pm jkX}.
$

Since the range of spatial frequencies is $ k\in[-\pi/X,\pi/X]$, we spontaneously have $ \vert c_k\vert\leq1$ for all $ k$. Therefore, we have shown by von Neumann analysis that the FDS Eq. (L.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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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]