A uniformly convergent scheme for two-parameter problems having layer behaviour. Issue 3 (4th March 2022)

• Record Type: Journal Article
• Authors: Kumar, Devendra
• Is Part Of: International journal of computer mathematics. Volume 99:Issue 3(2022)
• Page Start: 553
• Page End: 574
• Publication Date: 2022-03-04
• Subjects: Singularly perturbed problems -- two-parameter -- B-splines -- collocation method -- Shishkin mesh -- boundary layer; 65L10 -- 65L11 -- 65L20 -- 65L50 -- 65L70; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2021.1925115
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : We present a numerical scheme for the solution of two-parameter singularly perturbed problems whose solution has multi-scale behaviour in the sense that there are small regions where the solution changes very rapidly (known as layer regions) otherwise the solution is smooth (known as a regular region) throughout the domain of consideration. In particular, to solve the problems whose solution exhibits twin boundary layers at both endpoints of the domain of consideration, we propose a collocation method based on the quintic B -spline basis functions. A piecewise-uniform mesh that increases the density within the layer region compared to the outer region is used. An ( N + 1 ) × ( N + 1 ) penta-diagonal system of algebraic equations is obtained after the discretization. A well-known fast penta-diagonal system solver algorithm is used to solve the system. We have shown that the method is almost fourth-order parameters uniformly convergent. The theoretical estimates are verified through numerical simulations for two test problems.