Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions

TL;DR

This paper develops a computer algebra-based method to derive slow manifold dynamics for 2D lattice patches in reaction-diffusion systems, enhancing the equation-free approach with detailed algebraic procedures.

Contribution

It introduces a novel algebraic-computational framework for analyzing 2D patch dynamics within the equation-free methodology, addressing the complexity of algebraic detail involved.

Findings

Computer algebra procedures effectively handle complex 2D patch dynamics.

The methodology is adaptable to various reaction-diffusion equations.

A mixed numerical and algebraic approach is necessary for accurate modeling.

Abstract

Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. Here we explore the methodology applied to the gap-tooth scheme in the equation-free approach of Kevrekidis in two spatial dimensions. The algebraic detail is enormous so we detail computer algebra procedures to handle the enormity. However, modelling the dynamics on 2D spatial patches appears to require a mixed numerical and algebraic approach that is detailed in this report. Being based upon the computation of residuals, the procedures here may be simply adapted to a wide class of reaction-diffusion equations.