Taiwanese Journal of Mathematics

New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations

Xuefei He and Kun Wang

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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