Uniformly Convergent Second Order Numerical Method for a Class of Parameterized Singular Perturbation Problems
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Uniformly Convergent Second Order Numerical Method for a Class of Parameterized Singular Perturbation Problems D. Shakti1 · J. Mohapatra1
© Foundation for Scientific Research and Technological Innovation 2017
Abstract In this article, a class of nonlinear singularly perturbed boundary value problems depending on a parameter are considered. To solve this class of problems; first we apply the backward Euler finite difference scheme on Shishkin type meshes [standard Shishkin mesh (S-mesh), Bakhvalov–Shishkin mesh (B–S-mesh)]. The convergence analysis is carried out and the method is shown to be convergent with respect to the small parameter and is of almost first order accurate on S-mesh and first order accurate on B–S-mesh. Then, to improve the accuracy of the computed solution from almost first order to almost second order on S-mesh and from first order to second order on B–S-mesh, the post-processing method namely, the Richardson extrapolation technique is applied. The proof for the uniform convergence of the proposed method is carried out on both the meshes. Numerical experiments indicate the high accuracy of the proposed method. Keywords Parameterized problem · Boundary layer · Richardson extrapolation · Singular perturbation Mathematics Subject Classification 65L10 · 65L12
Introduction Singular perturbation problems (SPPs) are exceptionally useful because of its multi-character nature. It has vast application in the various field of science and applied mathematics. Many mathematical models starting from fluid dynamics to the problems in mathematical biology are modeled by SPPs. Typical examples include high Reynold’s number flow in the fluid dynamics, heat transport problem etc. For more details on singular perturbation, one can refer the books [5,8] and the references therein.
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J. Mohapatra [email protected] Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India
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Differ Equ Dyn Syst
Here, we consider the following singularly perturbed boundary value problem (BVP) depending on a parameter: εu (x) + f (x, u, λ) = 0, x ∈ Ω = (0, 1], (1) u(0) = s0 , u(1) = s1 , where 0 < ε 1 is small and known as the singular perturbation parameter, λ known as the control parameter and s0 , s1 are given constants. The function f (x, u, λ) is assumed to be sufficiently smooth and ⎧ f (x, u, λ) ∈ C 3 ([0, 1] × R2 ), ⎪ ⎪ ⎪ ⎪ ∂f ⎨ 0
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