Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems

TL;DR: It is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solutions or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution.

Abstract: This paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction-diffusion-convection equations under nonlinear boundary conditions. Three monotone iterative schemes for the finite difference system are presented, and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution. A comparison theorem for the various monotone sequences is given. It is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction-diffusion-convection equation as the mesh size decreases to zero. Some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained, and a discussion on the numerical stability of these schemes is given. Also given are some numerical results of a logistic reaction diffusion problem.