Finite Difference Schemes for Convection-Diffusion Problems

Abstract

This chapter is concerned with finite-difference discretisations of the stationary linear convection-diffusion problem

$$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1),\;\;u(0) = \gamma _0 ,\;u(1) = \gamma _1 ,$$

with b ≥ β > 0 on [0, 1]. For the sake of simplicity we shall assume that

$$c \ge 0\,\,{\rm and}\;\;{\rm b'} \ge 0\;\;\;\;\;{\rm on[0,1]}{\rm .}$$

Using (4.1) as a model problem, a general convergence theory for certain firstand second-order upwinded difference schemes on arbitrary and on layer-adapted meshes is derived. The close relationship between the differential operator and its upwinded discretisations is highlighted.