On solving a special class of weakly nonlinear finite difference systems

In this paper we consider the Newton–iterative method for solving weakly nonlinear finite difference systems of the form F (u) = Au + G(u) = 0, where the Jacobian matrix G′(u) satisfies an affine invariant Lipschitz condition. We also consider a modification of the method for which we can improve the likelihood of convergence from initial approximations that may be outside the attraction ball of the Newton–iterative method. We analyse the convergence of this damped method in the framework of the line search strategy. Numerical experiments on a diffusion–convection problem show the effectiveness of the method.