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Adelaide Research and Scholarship
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Schools and Disciplines
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School of Mathematical Sciences
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Mathematical Sciences Publications
Permanent link to this item:
http://hdl.handle.net/2440/49292
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| Type: | Technical report |
| Title: | Computer algebra derives the slow manifold of macroscale holistic discretisations in two dimensions |
| Author: | MacKenzie, Tony
Roberts, Andrew James |
| Issue Date: | 29-Mar-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 |
| SEO Classification: | 97 Expanding Knowledge
9701 Expanding Knowledge
970101 Expanding Knowledge in the Mathematical Sciences |
| Keywords: | dynamical systems; modelling; emergence; computer algebra; software |
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| Appears in Collections: | Mathematical Sciences Publications
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