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Type: Report
Title: Computer algebra derives the slow manifold of macroscale holistic discretisations in two dimensions
Author: MacKenzie, Tony
Roberts, Andrew James
Issue Date: 2009
School/Discipline: School of Mathematical Sciences : Applied Mathematics
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.
Contents: 1 Introduction 2 Construct the algebraic slow manifold 2.1 Initialisation 2.2 Iteratively construct the slow manifold 2.3 Scrounge an extra order of evolution using solvability 2.4 Derive the finite difference form 2.5 Obtain the equivalent PDE at full coupling 3 Numerically construct the slow manifold 3.1 Initialisation 3.2 Iteratively construct the slow manifold 3.3 LU decomposition 3.4 LU back substitution References
Keywords: dynamical systems; modelling; emergence; computer algebra; software
Appears in Collections:Mathematical Sciences publications

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