A second order accurate, positive scheme for singularly perturbed boundary value problems
type: Item
instance of: scholarly article
title: A second order accurate, positive scheme for singularly perturbed boundary value problems
zbMATH Open document ID: 0633.65076
DOI: 10.1007/BF00317059
published in: Computational Mechanics
publication date: 1988-01-01T00:00:00Z
review text: A new finite difference scheme of positive type is proposed for the numerical solution of singularly perturbed two-point boundary value problems of type: \(Lu=-\epsilon u''+f(x)u'+g(x)u=q(y)\), \(u(0)=a\), \(u(1)=b\), where \(f,g>0\), q are smooth functions and the small parameter \(\epsilon \ll O(h)\). The finite difference method given here may be obtained applying a standard central difference scheme to the following perturbed version of \((1)\quad L_ 0w=-\epsilon_ 0(x)w''+f_ 0(x)w'+g_ 0(x)w=q_ 0(x),\) where \(\epsilon_ 0(x)=\epsilon_ 0+\delta f^ 2\), \(f_ 0=f-\delta ff'-\delta fg,\) \(g_ 0=g-\delta fg',\) \(q_ 0=q-\delta fq',\) and \(\delta\) is a suitable function. It is proved that under suitable assumptions on \(\delta\) the global errors are of order \(O(h^ 2+\epsilon h)\) in regions which exclude a few mesh points in the neighborhood of possible layers. Some numerical experiments with several test problems proposed by \textit{C. E. Pearson} [J. Math. Phys. 47, 134-154 (1968; Zbl 0167.158)] are considered, but it is not clear how to choose \(\delta\) in an optimal way. Finally an extension of this scheme for 2-D problems is presented.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 35E15
zbMATH DE Number: 4030873
zbMATH Keywords: finite difference scheme of positive type
zbMATH Keywords: singularly perturbed two-point boundary value problems
zbMATH Keywords: small parameter
zbMATH Keywords: global errors
zbMATH Keywords: numerical experiments
zbMATH Keywords: test problems
author: V. J. Ervin
author: William Layton
MaRDI profile type: Publication
cites work: RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER
cites work: An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem
cites work: Q3853988
cites work: Numerical method for boundary layers with blowing - The exponential box scheme
cites work: Analysis of Some Difference Approximations for a Singular Perturbation Problem Without Turning Points
cites work: On central difference approximations to general second order elliptic equations
cites work: On a Differential Equation of Boundary Layer Type
full work available at URL: https://doi.org/10.1007/bf00317059
OpenAlex ID: W2012612587
description: scientific article; zbMATH DE number 4030873