A Fourth Order Compact Difference Scheme on Face Centered Cubic
Grids with Multigrid Method for Solving 2D Convection Diffusion Equation

Haiwei Sun, Ning Kang, 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

and
Eric S. Carlson
Department of Chemical Engineering
University of Alabama
P. O. Box 870203
Tuscaloosa, AL 35487-0203, USA

Abstract

We present a fourth order compact finite difference scheme on the face centered cubic (FCC) grids for the numerical solution of the two dimensional convection diffusion equation. The seven point formula is defined on a regular hexagon, where the strategy of directional derivative is employed to make the derivation procedure straightforward, efficient, and concise. A corresponding multigrid method is developed to solve the resulting sparse linear system. Numerical experiments are conducted to verify the fourth order convergence rate of the derived discretization scheme and to show that the fourth order compact difference scheme is computationally more efficient than the standard second order central difference scheme.


Key words: convection diffusion equation, multigrid method, face centered cubic grid, fourth order compact scheme

Mathematics Subject Classification: 65F10, 65N06, 65N22, 65N55, 76D07


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This paper has been published in Mathematics and Computers in Simulation, Vol. 63, No. 6, pp. 651--661 (2003).

Technical Report 341-02, 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 Japanese Research Organization for Information Science & Technology, and in part by the University of Kentucky Research Committee.