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Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern

A. J. Roberts
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Abstract

I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries.

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A. J. Roberts. 2001-10-15. Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern. https://arxiv.org/abs/math/0110153

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