Computer algebra derives the slow manifold of macroscale holistic discretisations in two dimensions

Abstract

Recent developments in dynamical systems theory provides new support for the discretisation of PDEs and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of the microscale dynamics of the PDE. Here we explore the methodology for problems with two spatial dimensions. The algebraic detail is enormous so we detail computer algebra procedures to handle the complexity. However, only low order models can be constructed purely algebraically; higher order models in 2D appear to require a mixed numerical and algebraic approach that is also detailed. Being based upon the computation of residuals, the procedures here may be simply adapted to a wide class of reaction-diffusion equations.