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Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem

Author: Eugene C. Gartland
Journal: Math. Comp. 48 (1987), 551-564
MSC: Primary 65L10; Secondary 34B05, 34E15
MathSciNet review: 878690
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Abstract: A family of uniformly accurate finite-difference schemes for the model problem $- \varepsilon u”+ a(x)u’+ b(x)u = f(x)$ is constructed using a general finite-difference framework of Lynch and Rice [Math. Comp., v. 34, 1980, pp. 333-372] and Doedel [SIAM J. Numer. Anal., v. 15, 1978, pp. 450-465], A scheme of order ${h^p}$ (uniform in $\varepsilon$) is constructed to be exact on a collection of functions of the type $\{ 1,x, \ldots ,{x^p},\exp (\frac {1}{\varepsilon }\smallint a),x\exp (\frac {1}{\varepsilon }\smallint a), \ldots ,{x^{p - 1}}\exp (\frac {1}{\varepsilon }\smallint a)\}$. The high order is achieved by using extra evaluations of the source term f. The details of the construction of such a scheme (for general p) and a complete discretization error analysis, which uses the stability results of Niederdrenk and Yserentant [Numer. Math., v. 41, 1983, pp. 223-253], are given. Numerical experiments exhibiting uniform orders ${h^p}$, $p = 1,2,3, \text {and}\;4$, are presented.

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Article copyright: © Copyright 1987 American Mathematical Society