A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer

• Main Authors: Zhongdi Cen, Anbo Le, Aimin Xu
• Format: Article
• Language: English
• Published: SpringerOpen 2017-07-01
• Series: Advances in Difference Equations
• Subjects: singular perturbation; reaction-diffusion equation; Shishkin-type mesh; finite difference scheme; uniform convergence
• Online Access: http://link.springer.com/article/10.1186/s13662-017-1268-1

Abstract In this paper, we consider a singularly perturbed reaction-diffusion problem with a discontinuous source term. Boundary and interior layers appear in the solution. The problem is discretized by using a hybrid finite difference scheme on a Shishkin-type mesh. A nonequidistant generalization of the Numerov scheme is used on the Shishkin-type mesh except for the point of discontinuity, whereas a second-order difference scheme with an additional refined mesh is used for the point of discontinuity. Although the difference scheme does not satisfy the discrete maximum principle, the maximum norm stability of the scheme is established. The maximum error in the mesh points is shown to be uniformly bounded by ( N − 1 ln N ) 4 $( N^{-1}\ln N ) ^{4}$ with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.