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2011 Adaptive Optimal m -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems
Jui-Ling Yu
J. Appl. Math. 2011(SI1): 1-25 (2011). DOI: 10.1155/2011/389207


We present a class of numerical methods for the reaction-diffusion-chemotaxis system which is significant for biological and chemistry pattern formation problems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. Along with the implementation of the method of lines, implicit or semi-implicit schemes are typical time stepping solvers to reduce the effect on time step constrains due to the stability condition. However, these two schemes are usually difficult to employ. In this paper, we propose an adaptive optimal time stepping strategy for the explicit $m$-stage Runge-Kutta method to solve reaction-diffusion-chemotaxis systems. Instead of relying on empirical approaches to control the time step size, variable time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.


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Jui-Ling Yu. "Adaptive Optimal m -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems." J. Appl. Math. 2011 (SI1) 1 - 25, 2011.


Published: 2011
First available in Project Euclid: 29 August 2011

zbMATH: 1220.65126
MathSciNet: MR2810573
Digital Object Identifier: 10.1155/2011/389207

Rights: Copyright © 2011 Hindawi


Vol.2011 • No. SI1 • 2011
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