A family of higher-order numerical schemes for the convection-diffusion equation in conservation form

Abstract

In this paper, we introduce a class of compact, higher-order, exponentially-fitted discretizations of the steady, one-dimensional convection-diffusion equation in conservation form. This will be achieved by employing the Mehrstellenverfahren technique of L. Collatz. These new schemes are shown to be uniformly second order accurate in the small parameter ε (the diffusion). The fourth-order accurate member of this family of schemes is studied in detail. Theoretical and numerical comparisons will be made to the methods of Allen-Southwell, El-Mistikawy-Werle, Leventhal, and Gartland. Additional examples exercising this new scheme are presented.