Convergence Proof of Jacobi Iterative Method for
A Discretized 2D Convection-Diffusion Equation

Deyu Sang
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

Abstract

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.