Holistic discretisation ensures fidelity to dynamics in two spatial dimensions

TL;DR

This paper develops a systematic, high-order discretisation method for reaction-diffusion PDEs in two dimensions, ensuring fidelity to the underlying dynamics through centre manifold theory and algebraic-computational techniques.

Contribution

It introduces a holistic discretisation approach that combines centre manifold theory and computer algebra to accurately model 2D reaction-diffusion systems at the macroscale.

Findings

Discretisations are consistent with PDEs to high order.

The method is systematically derived using computer algebra.

Higher order models require a mixed numerical and algebraic approach.

Abstract

Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of discrete models of the \pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the \pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.