Higher-order compact finite difference method for systems of reaction-diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction-diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme-substrate reaction-diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.