A surprising failure of the classical von Neumann analysis
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abstract
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.
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76th Annual Meeting of the Division of Fluid Dynamics
