Finite Volume Methods for Convection Diffusion Problems

Introduction In this paper we consider cell centered nite di erence approximations for second order convection di usion equations of divergence type Our goal is to construct nite di erence methods of second order of approximation that satisfy the discrete maximum principle The error estimates are in the discrete Sobolev spaces associated with the considered boundary value problem Approximation of the convection term in convection di usion problems by central nite di erences leads to schemes of second order which are stable only for su ciently small mesh size h The upwinding has been used to avoid the conditional stability but these approximations are of rst order and add substantial numerical di usion to the physical problem Various modi cations of the upwind schemes have been proposed aiming at a second order of accuracy and unconditional stability cf e g Samarskii see also Axelsson and Gustafson We investigate a number of modi ed upwind nite di erence strategies which provide both a second order of accuracy and that are unconditionally i e not only for small h stable There is a variety of techniques to derive and study nite di erence discretiza tions for di usion and convection di usion problems see e g Samarskii Ax elsson and Gustavson Spalding Il in etc In an error estimate of order O h in the discrete maximum norm for smooth solutions four continuous derivatives required is derived Another modi ed upwind nite di erence strategy leading to a second order scheme was considered in Axelsson and Gustafson Run chal and also Spalding have proposed and tested numerically upwind nite di erence schemes that can be used in both convection dominated and di usive lim its For one dimensional problems Il in has proposed nite di erence schemes for convection dominated second order equations and proved an O h error estimate in the maximum norm