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Development of finite difference schemes near an inflow boundary

Numerical schemes for a convection–diffusion problem defined on the whole real line have been derived by Morton and Sobey (IMA J. Numer. Anal. 1993; 13:141–160) using the exact evolution operator through one time step. In this paper we derive new numerical schemes by using the exact evolution operator for a convection–diffusion problem defined on the half‐line. We obtain a third‐order method that requires the use of a numerical boundary condition which is also derived using the same evolution operator. We determine whether there are advantages from the point of view of stability and accuracy in using these new schemes, when compared with similar methods obtained for the whole line. We conclude that the third‐order scheme provides gains in terms of stability and although it does not improve the practical accuracy of existing methods faraway from the inflow boundary, it does improve the accuracy next to the inflow boundary. Copyright © 2006 John Wiley & Sons, Ltd.