On Local Relaxation Methods and Their Application to Convection-Diffusion Equations

Abstract

This paper discusses local relaxation (LR) methods which can be regarded as generalizations of the successive overrelaxation (SOR) method. The difference is that within an LR method the relaxation factor is allowed to vary from equation to equation. A number of existing methods are found to be in fact special LR methods. Moreover, based on SOR theory, a new LR method is developed. The performance of LR methods is illustrated by applying them to central difference approximations of convection-diffusion equations. It is found that equations with small diffusion coefficients can be handled without difficulty. For equations with strongly varying coefficients, and for nonlinear equations, a properly selected LR method can be significantly more efficient than the optimum SOR method. As a special example, a 16 × 16 driven cavity problem for a Reynolds number of 10^6 can be solved in just a few seconds on a modern computer.