Journal of Applied Mathematics
Volume 2011 (2011), Article ID 389207, 25 pages
Research Article

Adaptive Optimal 𝑚 -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems

Department of Financial and Computational Mathematics, Providence University, Taichung 43301, Taiwan

Received 15 December 2010; Accepted 26 April 2011

Academic Editor: Shuyu Sun

Copyright © 2011 Jui-Ling Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.