A robust second-order numerical method for global solution and global normalized flux of singularly perturbed self-adjoint boundary-value problems

Abstract

In this paper, we consider the finite difference hybrid scheme constructed by Natesan et al. for obtaining uniformly convergent global solution and uniformly convergent normalized flux for self-adjoint singularly perturbed boundary value problems. The global solution is obtained from the numerical solution at the mesh points of this scheme, having almost second-order uniform convergence at the nodal points when it is constructed on a piecewise uniform Shishkin mesh. Using a classical cubic spline, we define the solution and the normalized flux on the entire domain. We prove that the uniform order of convergence of the global solution is the same as that of the hybrid scheme at the mesh points. In addition, the global normalized flux is also almost second-order uniformly convergent in the whole domain. We provide theoretical error bounds and some numerical examples showing the efficiency of the proposed technique for obtaining the global solution and the normalized flux.

Keywords:

• singular perturbation problems
• cubic spline
• reaction-diffusion problem
• Shishkin mesh
• global solution and global normalized flux

2000 AMS Subject Classifications :

• 65L10
• 65L12
• 65L20

Acknowledgements

This research was partially supported by the project MEC/FEDER MTM2007-63204 and the Diputación General de Aragón. The authors are thankful to the referees for their helpful comments in preparing the final version of this work.