A High Order Finite Difference Discretization Strategy
Based on Extrapolation for Convection Diffusion Equations

Haiwei Sun and Jun Zhang
Laboratory for High Performance Scientific Computing and Computer Simulation
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Abstract

We propose a new high order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth order accuracy coarse grid solution. A sixth order accuracy fine grid solution is obtained by interpolating the sixth order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth order combined compact difference (CCD) scheme, and the standard fourth order compact difference (FOC) scheme.

Key words: convection diffusion equation, compact difference scheme, CCD scheme, Richardson extrapolation.

Mathematics Subject Classification: 65N06, 65N55, 65F10.

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This paper has been published in Numerical Methods for Partial Differential Equations, Vol. 20, No. 1, pp. 18-32, 2004.

Technical Report 347-02, Department of Computer Science, University of Kentucky, Lexington, KY, 2002.

This research was supported in part by the U.S. National Science Foundation under the grant CCR-9902022, CCR-9988165, CCR-0092532, and ACI-0202934, in part by the U.S. Department of Energy under grant DE-FG02-02ER45961, in part by the Japan Research Organization for Information Science & Technology, and in part by the University of Kentucky Research Committee.