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Cen, Zhongdi; Le, Anbo; Xu, Aimin

A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer. (English) Zbl 1422.65130

Adv. Difference Equ. 2017, Paper No. 202, 14 p. (2017).
Summary: 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 \((N^{-1}\ln N)^{4}\) with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.

Cited in 3 Documents

MSC:

65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

singular perturbation; reaction-diffusion equation; Shishkin-type mesh; finite difference scheme; uniform convergence
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References:

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