A Two Colorable Fourth Order Compact Difference
Scheme and Parallel Iterative Solution of
the 3D Convection Diffusion Equation

Jun Zhang and Lixin Ge
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Jules Kouatchou
NASA Goddard Space Flight Center - Code 931
Greenbelt, MD 20771, USA


A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with a Gauss-Seidel type iterative method. This is compared with the known 19 point fourth order compact difference scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and 19 point fourth order compact schemes.

Key words: 3D convection diffusion equation, fourth order compact difference schemes, multigrid method, parallel computation.

Mathematics Subject Classification: 65M06, 65N12.

Technical Report No. 301-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. This research of the first two authors was supported in part by the U.S. National Science Foundation under the grant CCR-9902022, and in part by the University of Kentucky Center for Computational Sciences. This third author is also affiliated with Morgan State University and his research was supported by NASA under the grant No. NAGS-3508.