The Arithmetic Mean method for solving systems of nonlinear equations in finite differences

In this paper we consider the application of additive operator splitting methods for solving a finite difference nonlinear system of the form F(u) = (I - tau A(u))u - w = 0 generated by the discretization of two dimensional diffusion-convection problems with Neumann boundary conditions. Existence and uniqueness of a solution of this system has been proved under standard assumptions on the matrix A(u) and the source term w. Using the fact that the matrix A(u) can be decomposed into different splittings, we develop a nonlinear Arithmetic Mean method and a two-stage iterative method (a fixed-point-Arithmetic Mean method) for solving the system above. The convergence of these methods has been analyzed. Numerical experiments show that the fixed-point-Arithmetic Mean method is rapidly convergent when the diffusion coefficient is weakly nonlinear. (c) 2006 Elsevier Inc. All rights reserved.