Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion problems

Abstract

In this paper, we present an optimal fourth-order parameter-uniform non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion boundary value problems exhibiting boundary layers at both ends of the domain. We discretize the problem using a high-order non-monotone finite difference scheme and prove that the scheme is stable in the maximum norm. The equidistribution of an appropriate monitor function is used to generate the layer-adapted meshes to discretize the problem. The method is proved to be optimal fourth-order uniformly convergent on these equidistributed meshes. Numerical results are presented to validate the theory and to demonstrate the efficiency of the proposed method.

Keywords:

• Boundary layer
• adaptive meshes
• equidistribution principle
• optimal order convergence
• high-order scheme

2020 AMS Subject Classifications:

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

Disclosure statement

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

Additional information

Funding

This research was supported by the Science and Engineering Research Board (SERB) under Project No. ECR/2017/000564. The first author gratefully acknowledges the support of University Grant Commission, India, for research fellowship with reference No.: 20/12/2015(ii)EU-V.