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

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 with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented. MSC: 65L10; 65L12

The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

“…Compared with the existing difference schemes [9,18], the RSC-SSM accuracy has been significantly improved for the case with ε 2 ≤ 10 − 3 . However, when the parameters ε 2 ≥ 10 − 2 , the accuracy of the RSC-SSM is similar to that of the difference method.…”

“…e RSC-SSM method and the different schemes in [9,18] are used to solve this problem with several choices of N and ε. e maximum relative errors of the three methods are listed in Table 1.…”

“…A uniformly numerical convergence scheme based on piecewiseuniform Shishkin's meshes was developed for a reaction-diffusion equation with a discontinuous diffusion coefficient and proved to have second-order convergence [8]. A hybrid difference scheme on a Shishkin's mesh was introduced for a singularly perturbed reaction-diffusion problem with one and two parameters [9] with interior layers.…”

The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

“…By taking into account the finite element method, sign-changing solutions are obtained [16]. The Hybrid finite difference scheme and Numerov scheme are established on a Shishkin type mesh [17]. By carrying out boundary layer analysis, numerical solutions are obtained in a quasi--uniform mesh [18].…”

Absolutely stable difference scheme for a general class of singular perturbation problems

“…The stability analysis of the method is discussed showing that the method is absolutely stable under a certain condition on the deviating argument whereas there is no stability restriction on the step-size. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method [48,50] in addition to the absolute stability property [19,28,[48][49][50]. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method.…”