Spurious Behaviour of Numerically Computed Fluid Flow

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Abstract
We investigate the stability to aliasing errors of numerical schemes for hydrodynamics, taking the viscous Burgers' equation as a model for systems with a term that is quadratic in the velocity. Considering wavelengths equal to three times the mesh-spacing, and arbitrary mean flow, we are able to demonstrate explicitly for common schemes (a) a sufficient criterion for stability and (b) blow-up of solutions in a finite time when (a) is violated. Singular behaviour is shown to persist at all wavelengths: studies of wavelengths up to thirty times mesh-spacing make it clear that a profile with a single region of strong convergent flow is most conducive to instability. In contrast, spectral (Galerkin) and upwind schemes are shown to be stable for all flows and periods.

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