An almost second order uniformly convergent scheme for a singularly perturbed initial value problem

Cen Z., Erdoğan F., XU A.

NUMERICAL ALGORITHMS, vol.67, no.2, pp.457-476, 2014 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 67 Issue: 2
  • Publication Date: 2014
  • Doi Number: 10.1007/s11075-013-9801-0
  • Journal Name: NUMERICAL ALGORITHMS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.457-476
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

In this paper a nonlinear singularly perturbed initial problem is considered. The behavior of the exact solution and its derivatives is analyzed, 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. It is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameter. Numerical experiment supports these theoretical results.

In this paper a nonlinear singularly perturbed initial problem is considered. The behavior of the exact solution and its derivatives is analyzed, 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. It is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameter. Numerical experiment supports these theoretical results.