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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem
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by Eugene C. Gartland PDF
Math. Comp. 48 (1987), 551-564 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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