A Discretized 2D Convection-Diffusion Equation

Department of System Engineering and Applied Mathematics

Chongqing University

Chongqing 400044, P. R. China

Jun Zhang

Department of Computer Science

University of Kentucky

773 Anderson Hall

Lexington, KY 40506--0046, USA

Shiqing Zhang

Department of System Engineering and Applied Mathematics

Chongqing University

Chongqing 400044, P. R. China

We prove that the Jacobi iterative method converges from any initial values for solving the linear system resulting from a fourth-order compact finite difference discretization of the 2D convection-diffusion equation with constant convection coefficients. The convergence is assured regardless of the magnitude of the convection coefficients and the discretization mesh. This proof confirms the unconditional stability (convergence) of the 2D fourth-order compact scheme (with respect to classical iterative methods), a fact that has only been verified numerically but evaded rigorous justification for almost two decades.

Key words: Jacobi iterative method, fourth-order compact discretization scheme, convection-diffusion equation.

Technical Report No. 276-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998. This research is supported in part by the University of Kentucky Center for Computational Sciences.