Finite Difference Approximation of a Convection Diffusion Equation Near a Boundary

Abstract

Techniques for solving evolutionary convection diffusion equations are almost universally based on analysis of infinite domain situations. Almost all practical problems involve physical domains with boundaries. For a number of numerical schemes with Dirichlet boundary conditions, the numerical algorithm can be used without alteration near a boundary. The development of higher order methods such as Quickest or second order upwinding (including many schemes with flux limiters) can introduce difficulty near an inflow boundary, since for points adjacent to the boundary there are insufficient upstream points for the high order scheme to be applied without alteration. Usually reliance is placed on ad-hoc solutions for individual problems. Recently Morton & Sobey (1993) showed how analytic evolutionary solutions could be used to derive arbitrary accuracy finite difference and finite element schemes for constant coefficient convection diffusion. In this paper we continue that work by considering analytic solutions for evolution on the half line $x \leq 0$. We use an exact evolutionary operator to derive finite difference approximation schemes which maintain accuracy near a boundary. These schemes are applied and compared for a simple test problem which has an exact solution. As might be expected, it is not just accuracy but also stability, which dominates solution of time evolution problems.\ud \ud The work reported here forms part of the research programme of the Oxford-Reading Institute for Computational Fluid Dynamics