The exponential accuracy of Fourier and Chebyshev differencing methods
SIAM Journal on Numerical Analysis
The algorithmic beauty of sea shells
The algorithmic beauty of sea shells
Numerical solution of partial differential equations
Numerical solution of partial differential equations
An r-adaptive finite element method based upon moving mesh PDEs
Journal of Computational Physics
An adaptive mesh algorithm for evolving surfaces: simulation of drop breakup and coalescence
Journal of Computational Physics
Exponential time differencing for stiff systems
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A moving grid finite element method applied to a model biological pattern generator
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Journal of Scientific Computing
Journal of Computational Physics
Asymptotic Profile of Species Migrating on a Growing Habitat
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Computer Methods and Programs in Biomedicine
Numerical approximation of Turing patterns in electrodeposition by ADI methods
Journal of Computational and Applied Mathematics
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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 ADI 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.