High order generalized upwind schemes and numerical solution of singular perturbation problems

High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods. AMS subject classification (2000)  65L10, 65L12, 65L50