Taiwanese Journal of Mathematics

New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations

Xuefei He and Kun Wang

Full-text: Open access

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.

Article information

Source
Taiwanese J. Math., Volume 22, Number 4 (2018), 949-978.

Dates
Received: 22 March 2017
Revised: 2 October 2017
Accepted: 16 October 2017
First available in Project Euclid: 26 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1508983229

Digital Object Identifier
doi:10.11650/tjm/171002

Mathematical Reviews number (MathSciNet)
MR3830829

Zentralblatt MATH identifier
06965405

Subjects
Primary: 65L11: Singularly perturbed problems 65N06: Finite difference methods 65N15: Error bounds

Keywords
convection-diffusion equation finite difference method singularly perturbed problem Shishkin mesh error estimate

Citation

He, Xuefei; Wang, Kun. New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations. Taiwanese J. Math. 22 (2018), no. 4, 949--978. doi:10.11650/tjm/171002. https://projecteuclid.org/euclid.twjm/1508983229


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