A surprising failure of the classical von Neumann analysis

The von Neumann stability analysis is a classical widely used tool

to assess stability of numerical schemes for PDEs.

In this analysis, the so-called amplification factor,

defined as the ratio of the amplitude of the error at

two successive steps, must be bounded by unity in order

for the scheme to be stable.

Using a simple leap-frog scheme for the wave equation,

we show how the von Neumann analysis gives, surprisingly,

incorrect results. According to the analysis,

the amplification factor approaches a constant less than

or equal to unity when parameters are inside the stability

region. We show this is actually not the case.

The surprising failure of the approach, even for very simple cases,

can be traced to an implicit assumption in the

analysis of multi-step methods

which we investigate analytically. We show how and when

the von Neumann analysis gives correct results and when

alternative approaches are necessary.

We generalize the approach and provide an alternative

stability criterion that applies more broadly to

general multi-step methods.