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

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878690-0

MathSciNet review:
878690

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Abstract | References | Similar Articles | Additional Information

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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*Soviet Math. Dokl.*, v. 25, 1982, pp. 168-172.

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Article copyright:
© Copyright 1987
American Mathematical Society