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Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains. (English) Zbl 1122.65076
Summary: Reaction-diffusion systems have been widely studied in developmental biology, chemistry and, more recently, in financial mathematics. Most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions. It therefore becomes convenient to look for numerical solutions: finite difference, finite element, finite volume and spectral methods are typical examples of the numerical methods used. Most of these methods are locally based schemes. We examine the implications of mesh structure on numerically computed solutions of a well-studied reaction-diffusion model system on two-dimensional fixed and growing domains. The incorporation of domain growth creates an additional parameter – the grid-point velocity – and this greatly influences the selection of certain symmetric solutions for the alternating direction implicit finite difference scheme when a uniform square mesh structure is used. Domain growth coupled with grid-point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure. We compare our results to those obtained by use of finite elements on unstructured triangular elements.
MSC:
65M06Finite difference methods (IVP of PDE)
35K57Reaction-diffusion equations
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)