A uniformly convergent scheme for two-parameter problems having layer behaviour

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 $\mathcal{B}$-spline basis functions. A piecewise-uniform mesh that increases the density within the layer region compared to the outer region is used. An $(N+1)\times (N+1)$ 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.

Keywords:

• Singularly perturbed problems
• two-parameter
• $\mathcal{B}$-splines
• collocation method
• Shishkin mesh
• boundary layer

AMS subject classifications:

• 65L10
• 65L11
• 65L20
• 65L50
• 65L70

Acknowledgments

The author is thankful to the anonymous reviewers for their careful reading of our manuscript and their valuable comments/suggestions which improved the organization and the quality of the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).