Stable high order methods for elliptic equations with large first order terms

Abstract

Stable high-order methods are developed for solving elliptic partial differential equations with large first order terms. In particular, second-order and fourth-order methods are developed by stabilizing the central difference method. The methods can solve second-order linear elliptic partial differential equations and are found to be stable and accurate in all tested examples. Results are presented for the two-dimensional convection-diffusion equation involving problems with and without boundary layers. The methods converged for all values of parameters attempted and the results compared favorably with other methods. The methods developed in this paper are accurate, stable, easy to use and applicable to other problems.