Nonlinear Delay Reaction Diffusion Equations, Finite Difference Method

Many models in various field of applications, such as biology, ecology and biochemistry, can be described by nonlinear reaction-diffusion equations with time delays. It is of practical interest to give an efficient numerical method for solving the equations. In this paper, a finite difference method with high accuracy is established for a class of nonlinear reaction-diffusion equations with time delays. This method has second order accuracy in time and fourth order accuracy in space. Some qualitative analyses are given for nonlinear finite difference schemes. This includes existence-uniqueness of finite difference solutions and the convergence of the finite difference solution to a continuous solution. To solve the nonlinear finite difference scheme, an accelerated monotone iterative method is presented, and the explicit estimate for the rate of convergence is given. The rate of convergence is nearly quadratic, and is quadratic under some additional conditions. The numerical results show the advantages of the method.