A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

Authors:
Yuan-Ming Wang;Jie Wang
Affiliations:
Department of Mathematics, East China Normal University, Shanghai 200241, People's Republic of China and Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Sci ...;School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, People's Republic of China
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.