Algorithms composition approach based on difference potentials method for parabolic problems

In this work we develop an efficient and flexible Algorithms Composition Approach based on the idea of the difference potentials method (DPM) for parabolic problems in composite and complex domains. Here, the parabolic equation serves both as the simplified model, and as the first step towards future development of the proposed framework for more realistic systems of materials, fluids, or chemicals with different properties in the different domains. Some examples of such models include the ocean-atmosphere models, chemotaxis models in biology, and blood flow models. Very often, such models are heterogeneous systems—described by different types of partial differential equations (PDEs) in different domains—and must take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient and flexible numerical method that can capture certain properties of analytical solutions in different domains, while handling the arbitrary geometries and complex structures of the subdomains. The Algorithms Compositions principle, as well as the Domain Decomposition idea, is one way to overcome these difficulties while developing very efficient and accurate numerical schemes for the problems. The Algorithms Composition Approach proposed here can handle the complex geometries of the domains without the use of unstructured meshes, and can be employed with fast Poisson solvers. Our method combines the simplicity of the finite difference methods on Cartesian meshes with the flexibility of the Difference Potentials method. The developed method is very well suited for parallel computations as well, since most of the computations in each domain are performed independently of the others.

Keywords

parabolic equation, heat equation, Calderon’s potentials, Calderon’s boundary equations with projections, difference potentials methods, finite difference, Cartesian meshes, complex and composite domains, curvilinear boundaries, algorithms composition, and domain decomposition

2010 Mathematics Subject Classification

35K05, 35K20, 65M06, 65M22, 65M55, 65M70

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Published 7 February 2014