Quick von Neumann stability analysis

von Neumann stability analysis is a great tool to assess the stabilty of discretizing schemes for PDEs. But too often, imho, the discussion is too convoluted. Here, I try to provide a shortcut.

In a nutshell:

  • The standard approach involves Fourier techniques, involving (of course) complex numbers
  • The real part of these numbers is analysed, with some trigonometric expression resulting, identifying the troublesome modes
  • I claim this mode can be identified in advance, which makes the whole Fourier procedure unnecessary

BTW: it’s pronounced “fon no ee man”

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Quick and dirty Rayleigh-Taylor instability

The Rayleigh-Taylor instability is a well-known benchmarck for CFD codes. The idea is to start with two phases, on on top of the other, the lighter one being underneath. The interface is slightly perturbed, and this plume appears. I describe a quick and dirty way of getting this instability.

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