Resonances Introduced by Discretization

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Abstract
This paper examines the effect of spatially discretizing the non-linear equation chosen to model the integro-differential equations governing interfacial vortex sheets. Here H is the Hilbert transform and 〈 〉 is a spatial average. It is shown numerically and analytically that small-amplitude travelling-wave solutions of the discretized model equation are subject to instabilities whose growth rate is proportional to the amplitude and depends in a complicated way on the number of mesh points per wavelength. The mechanism of the instability is shown to be a resonance not present in the continuous system and present in the discrete system because the frequency of waves with wavelength equal to twice the mesh spacing is zero.

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