Modelling the evolution of ideal, infinite domain patterns, on a finite domain using a Perfectly Matched Layer

Abstract

The Swift-Hohenberg equation is an evolution equation which can produce a Pattern, or a pattern-like picture, to be more precise. For example, it could be used to model some simple natural patterns, like stripes and rolls that one may observe in a Rayleigh-Benard convection experiment.

But for any pattern formation obtained by an evolution equation to look ideal, we have to consider this equation analytically on the infinite domain. If one wants to calculate and present the results numerically, the problem has to be discretized, the number of steps turns out to be finite then, and at some point the lateral boundaries appear. These boundaries cause a backward reflection and destroy the pattern eventually.

The idea of using a Perfectly Matched Layer in order to obtain some reflectionless boundaries was suggested. It should let us model the evolution of an ideal, infinite domain pattern, on an actual finite domain. From a set of numerical methods the most suitable ones will be chosen. The MATLAB environment is used to write a code to visualise the results.

Any investigations that could concern the fact of applying a Perfectly Matched Layer to an evolution Swift-Hohenberg equation seem to be absolutely new and yet untouched. This Thesis might be considered as a first step, as an introduction to the problem and its possible solutions.