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An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems. (English) Zbl 1126.65085

Summary: Reaction-diffusion-chemotaxis systems have proven to be fairly accurate mathematical models for many pattern formation problems in chemistry and biology. These systems are important for computer simulations of patterns, parameter estimations as well as analysis of biological systems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations.
In this paper, a general reaction-diffusion-chemotaxis system is considered for specific numerical issues of pattern simulations. We propose a fully explicit discretization combined with a variable optimal time step strategy for solving the reaction-diffusion-chemotaxis system. Theorems about stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Results of a numerical experiment on a model problem are given for comparison with other numerical methods. Simulations on two real biological experiments are also given.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65L12 Finite difference and finite volume methods for ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
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