Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions

Kopteva, Natalia; Stynes, Martin

doi:10.1016/j.apnum.2004.07.001

Cited by 37 publications

(39 citation statements)

References 11 publications

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“…In this section we present numerical results to confirm the uniform accuracy of the discrete problem (12)- (14). To demonstrate the efficiency of the method, we present two examples having boundary layers.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In this section we prove the theorem on ϵ-uniform convergence of the discrete problem (12)- (14). The proof uses the decomposition of the solution y to the problem (1)-(2) to the layer s and a regular component r given by …”

Section: Uniform Convergencementioning

confidence: 99%

“…Various different difference schemes have been constructed which are uniformly convergent on equidistant meshes as well as schemes on specially constructed, mostly Shishkin and Bakvhvalov-type meshes, where ϵ-uniform convergence of second order has been demonstrated, see e.g. [11,13,14,24,26,28,29], as well as schemes with ϵ-uniform convergence of order greater than two, see e.g. [7,8,9,32,33].…”

Section: Introductionmentioning

confidence: 99%

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A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

Pasic

Enes

Karasuljić

et al. 2015

J. of Mod. Meth. in Numer. Math.

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We are considering a semilinear singular perturbation reaction -diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is ϵ-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Uniform Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

Pasic

Enes

Karasuljić

et al. 2015

J. of Mod. Meth. in Numer. Math.

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“…Conditions (A1), (A2) intrinsically arise from the asymptotic analysis of problem (1.1) and guarantee that there exists a boundary-layer solution u of (1.1) such that u ≈ u 0 in the interior subdomain of Ω away from the boundary, while the boundary layer is of width O(ε| ln ε|) [6,12]; see Theorem 2.2 for a precise statement and [8] for a detailed discussion of (A1), (A2) in one dimension. Note that assumption (A1) is local, i.e., the reduced problem (1.2) is permitted to have more than one stable solution.…”

Section: Note That If G(x)mentioning

confidence: 99%

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

Kopteva¹

2006

Math. Comp.

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Abstract. A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε 2 is arbitrarily small, which induces boundary layers. Constructing discrete sub-and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in ε for ε ≤ Ch. Here h > 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed Ch −2 . Numerical experiments are performed to support the theoretical results.

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“…While the analysis and numerical results in this paper are given for the one-dimensional problem (1.1), much of what is here can be generalized to analogues of (1.1) posed in higher dimensions; compare the one-dimensional nonlinear problem discussed in [10] and the extension of this work to the two-dimensional case in [8], where a theoretical analysis and numerical results are presented. The onedimensional analysis is already so complex that the extra notation required to explain it in two dimensions would only obscure the central ideas that we wish to communicate.…”

Section: Introductionmentioning

confidence: 99%

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Kopteva

Stynes

2011

Numer. Math.

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Abstract.A semilinear reaction-diffusion two-point boundary value problem, whose secondorder derivative is multiplied by a small positive parameter ε 2 , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult.To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub-and super-solutions. Convergence results are deduced that depend on the relative sizes of ε and N , where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton's method to compute a discrete solution are discussed.

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Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions

Cited by 37 publications

References 11 publications

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

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