Rational Modelling Determines Boundary Conditions
by
A. J. Roberts
University of Southern Queensland

Models of spatio-temporal dynamics, such as the Kuramoto-Sivashinsky equation and the Ginzburg-Landau equation, require boundary conditions at the edge of the spatial domain. Traditional derivations of boundary conditions for such models arguably have weak foundations. First, we investigate deriving discretisations near boundaries using the attractive centre manifold theory. This approach goes some way to ensuring models are valid for finite grid spacing by accounting for subgrid scale processes. Such multiscale modelling is a key issue in many numerical models and is usually addressed heuristically. Second, the we consider the evolution near boundaries of the large-scale solutions of models such as the Kuramoto-Sivashinsky equation. By interchanging the roles of space and time, centre manifold theory resolves how boundary layers resolve into the smooth interior solutions. This resolution then provides boundary conditions for the models-in principle because some of the details are very complicated. This generic approach is a direction for development of a rational approach to boundary conditions.

Date received: September 26, 2003