New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations

TLDR: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered, which can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter.

Abstract: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.