A Monotone Compact ADI Method For Systems Of Reaction-Diffusion Equations

This paper is concerned with a compact ADI method for solving systems of two-dimensional reaction-diffusion equations. This method has the accuracy of fourth-order in space and second-order in time. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is provided for solving the resulting discrete system, and the sequences of iterations converge monotonically to a unique solution of the system. Numerical results demonstrate the high efficiency and advantages of this new approach.