A comparison between MMAE and SCEM for solving singularly perturbed linear boundary layer problems

TL;DR

This paper introduces the Successive Complementary Expansion Method (SCEM), an asymptotic approach for singularly perturbed boundary problems that simplifies the process and outperforms the traditional Method of Matched Asymptotic Expansions (MMAE).

Contribution

The paper presents SCEM, a new asymptotic method that avoids the matching process of MMAE, providing a more straightforward and robust solution for linear boundary layer problems.

Findings

SCEM yields uniformly valid approximations without matching.

Numerical experiments show SCEM's superior accuracy and robustness.

SCEM outperforms MMAE in solving linear boundary layer problems.

Abstract

In this study, we propose an efficient method so-called Successive Complementary Expansion Method (SCEM), that is based on generalized asymptotic expansions, for approximating to the solutions of singularly perturbed two-point boundary value problems. In this easy-applicable asymptotic method, in contrast to the well-known method the Method of Matched Asymptotic Expansions (MMAE), the matching process is not necessary to obtain uniformly valid approximations. The key point: A uniformly valid approximation is adopted first, and complementary functions are obtained imposing the corresponding boundary conditions. An illustrative and two numerical experiments are provided to show the implementation and numerical properties of the present method. Furthermore, MMAE results are also given in order to compare the numerical robustness of the methods. Numerical results and the comparisons demonstrate absolute superiority of SCEM over MMAE for linear problems.