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):


where





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



![$\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..
$](http://www.dsprelated.com/josimages_new/pasp/img4501.png)


![$ k\in[-\pi/X,\pi/X]$](http://www.dsprelated.com/josimages_new/pasp/img4504.png)



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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