A Finite Difference Method For Time-Periodic Solutions Of Nonlinear Parabolic Systems

Time-periodic behavior of solutions arises from many problems in various field of applied sciences, such as biology, ecology, biochemistry and physics, and many of these phenomena are usually described by a coupled system of nonlinear parabolic equations. It is of certain practical interests to give an efficient numerical method for such systems. In this paper, a finite difference method with high accuracy is established for solving time-periodic solutions of a class of nonlinear parabolic systems in one-dimensional domain. This method has second order accuracy in time and fourth order accuracy in space. Some qualitative analyses are given for the nonlinear finite difference scheme. This includes the existence-uniqueness of finite difference solution and the convergence of the finite difference solution to the analytical solution. To solve the nonlinear finite difference scheme, an efficient monotone iterative algorithm is developed. The sequences of iterations converge monotonically to an unique solution of nonlinear finite difference system, and the initial iteration can be explicitly constructed without any knowledge of the solution. The numerical results demonstrate the advantages of the method, including the monotone convergence property of iterative sequences and the high accuracy of the method.