We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19-point fourth order discretization scheme to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equation with a high Reynolds number, we also implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact scheme (SOC), compared with the previously published fourth order compact scheme (FOC).

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

Technical Report CMIDA-HiPSCCS 016-09, Department of Computer Science, University of Kentucky, Lexington, KY, September 23, 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.