Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem

Abstract

A periodic boundary value problem with a small parameter multiplying the first- and second-order derivatives is considered. The problem is discretized using a hybrid difference scheme on a Shishkin mesh. We show that the scheme is almost second-order convergent in the maximum norm, which is independent of a singular perturbation parameter. Numerical experiment supports these theoretical results.

Keywords:

• singular perturbation
• periodical boundary value problem
• hybrid difference scheme
• Shishkin mesh
• uniform convergence

2000 AMS Subject Classifications :

• Primary: 65L10
• Secondary: 65L12
• 65L50

Acknowledgement

I would like to thank the anonymous referees for several suggestions for the improvement of this paper. The work was supported by Zhejiang Province Natural Science Foundation (Grant No.Y607504) of China and Ningbo Natural Science Foundation (Grant No. 2009A610082) of China.