A uniformly convergent hybrid difference scheme for a system of singularly perturbed initial value problems. Issue 5 (3rd May 2020)

• Record Type: Journal Article
• Authors: Cen, Zhongdi; Xu, Aimin; Le, Anbo; Liu, Li-Bin
• Is Part Of: International journal of computer mathematics. Volume 97:Issue 5(2020)
• Page Start: 1058
• Page End: 1086
• Publication Date: 2020-05-03
• Subjects: Singular perturbation -- initial value problem -- finite difference scheme -- Shishkin mesh -- uniform convergence; Primary: 65L10 -- Secondary: 65L12 -- 65L50; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2019.1605063
• Languages: English
• ISSNs: 0020-7160

Abstract

ABSTRACT: A system of second-order singularly perturbed initial value problems with weak hypotheses on the coefficients is considered. The equations have diffusion parameters of different magnitudes, which gives rise to overlapping boundary layers. The structure of these layers is analysed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second-order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall's inequality, it is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameters. Numerical experiments are provided to validate the theoretical results.