We present an explicit sixth order compact finite difference scheme for fast high accuracy numerical solutions of the two dimensional convection diffusion equation with variable coefficients. The sixth order scheme is based on the well-known fourth order compact scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine-point fourth order compact scheme and Sun-Zhang's sixth order method. Two convection diffusion problems are solved numerically to validate our proposed sixth order scheme.
Mathematics Subject Classification: 65N06, 65N55, 65F10.
Technical Report CMIDA-HiPSCCS 015-09, Department of Computer Science, University of Kentucky, Lexington, KY, 2009.
The research work of Dr. Jun Zhang was supported in part by NSF under grant CCF-0727600, in part by the Kentucky Science and Engineering Foundation under grant KSEF-148-502-06-186, and in part by NIH under grant 1 R01 HL086644-01.