High order fitted operator numerical method for self-adjoint singular perturbation problems
type: Item
instance of: scholarly article
title: High order fitted operator numerical method for self-adjoint singular perturbation problems
zbMATH Open document ID: 1086.65079
author: Kailash C. Patidar
published in: Applied Mathematics and Computation
publication date: 2006-02-07T00:00:00Z
review text: A numerical scheme is proposed for self-adjoint singular perturbation problems in conservation form \[ -\varepsilon (a(x)y')' + b(x)y = f(x), \quad x \in [0,1] \] subject to nonhomogeneous Dirichlet boundary conditions, for sufficiently smooth data satisfying \(a(x) \geq a > 0, b(x) \geq b >0\), and \(0 < \varepsilon \ll 1\). The same problem has been numerically solved by using a cubic spline difference scheme with exponentially fitting factor \textit{M. K. Kadalbajoo} and \textit{K. C. Patidar} [Int. J. Math. Math. Sci. 2003, No. 61, 3873--3891 (2003; Zbl 1045.65064)]. The same trick is adapted here just by replacing the cubic spline scheme by the Numerov scheme with the same fitting factor. The author is claiming that the present scheme is of fourth-order accurate for moderate values of the singular perturbation parameter \(\varepsilon\), and for very smaller values of \(\varepsilon\) the method is \(\varepsilon\)-uniform convergent with order two. But this is not clear from the theorem on the error estimate. Indeed, the standard Numerov's scheme (i.e., without any fitting factor) produces good approximate results on a uniform mesh for moderate values of \(\varepsilon\). From the numerical tables on the convergence rate it is very well clear that the method is not fourth-order convergent. Also, one can notice that for certain values of \(\varepsilon\) the rate convergence will be less than one, but there is no explanation about these strange behavior.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 65L70
zbMATH DE Number: 5004374
zbMATH Keywords: singular perturbation
zbMATH Keywords: boundary value problems
zbMATH Keywords: exponentially fitted difference scheme, Numerov's method
zbMATH Keywords: error bound
reviewed by: Srinivasan Natesan
MaRDI profile type: Publication
full work available at URL: j.amc.2005.01.069
OpenAlex ID: W1971565872
cites work: RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER
cites work: A variational difference scheme for a boundary value problem with a small parameter in the highest derivative
cites work: On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type
cites work: Nonlinear singular perturbation phenomena: theory and applications
cites work: Q3909948
cites work: Q4485828
cites work: Q3321316
cites work: A survey of numerical techniques for solving singularly perturbed ordinary differential equations
cites work: Singularly perturbed problems in partial differential equations: A survey
cites work: Perturbation methods in applied mathematics
cites work: Multiple scale and singular perturbation methods
cites work: Q5674335
cites work: Efficient Integration Methods for Stiff Systems of Ordinary Differential Equations
cites work: Q3853070
cites work: Q4486278
cites work: Q4865485
cites work: Perturbation methods and non-linear hyperbolic waves
cites work: Q3895818
cites work: Q3943930
cites work: Q4774844
cites work: Singular perturbation methods for ordinary differential equations
cites work: A Uniformly Accurate Finite-Element Method for a Singularly Perturbed One-Dimensional Reaction-Diffusion Problem
cites work: Global uniformly convergent schemes for a singularly perturbed boundary- value problem using patched base spline-functions
cites work: Q4871736
cites work: On the Finite Element Method for Singularly Perturbed Reaction-Diffusion Problems in Two and One Dimensions
cites work: A uniformly accurate spline collocation method for a singular perturbation problem
cites work: Exponential splines difference scheme for singular perturbation problem with mixed boundary conditions
cites work: A lower bound for the smallest singular value of a matrix
cites work: Q5342712
cites work: Q4378275
DOI: 10.1016/J.AMC.2005.01.069
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description: scientific article; zbMATH DE number 5004374