Uniform highorder difference schemes for a singularly perturbed twopoint boundary value problem
Author:
Eugene C. Gartland
Journal:
Math. Comp. 48 (1987), 551564, S5
MSC:
Primary 65L10; Secondary 34B05, 34E15
MathSciNet review:
878690
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Abstract: A family of uniformly accurate finitedifference schemes for the model problem is constructed using a general finitedifference framework of Lynch and Rice [Math. Comp., v. 34, 1980, pp. 333372] and Doedel [SIAM J. Numer. Anal., v. 15, 1978, pp. 450465], A scheme of order (uniform in ) is constructed to be exact on a collection of functions of the type . 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. 223253], are given. Numerical experiments exhibiting uniform orders , and, are presented.
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 [1]
 L. R. Abrahamsson, H. B. Keller & H. O. Kreiss, "Difference approximations for singular perturbations of systems of ordinary differential equations," Numer. Math., v. 22, 1974, pp. 367391. MR 0388784 (52:9618)
 [2]
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 [3]
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 [4]
 U. Ascher & R. Weiss, "Collocation for singular perturbation problems I: First order systems with constant coefficients," SIAM J. Numer. Anal., v. 20, 1983, pp. 537557. MR 701095 (85a:65113)
 [5]
 U. Ascher & R. Weiss, "Collocation for singular perturbation problems. II: Linear first order systems without turning points," Math. Comp., v. 43, 1984, pp. 157187. MR 744929 (86g:65138a)
 [6]
 A. O. H. Axelsson, "Stability and error estimates of Galerkin finiteelement approximations for convectiondiffusion equations," IMA J. Numer. Anal., v. 1, 1981, pp. 329345. MR 641313 (83a:65105)
 [7]
 A. E. Berger, J. M. Solomon, M. Ciment, S. H. Leventhal & B. C. Weinberg, "Generalized operator compact implicit schemes for boundary layer problems," Math. Comp., v. 35, 1980, pp. 695731. MR 572850 (81f:65057)
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 J. E. Flaherty & W. Mathon, "Collocation with polynomial and tension splines for singularly perturbed boundary value problems," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 260289. MR 594760 (82a:65055)
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 S. H. Leventhal, "An operator compact implicit method of exponential type," J. Comput. Phys., v. 46, 1982, pp. 138165. MR 665807 (84b:76007)
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 J. J. H. Miller (Editor), Proc. BAIL III Conf., Boole Press, Dublin, 1984. MR 774603 (85k:00008)
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 K. Niederdrenk & H. Yserentant, "Die gleichmässige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme," Numer. Math., v. 41, 1983, pp. 223253. MR 703123 (84j:65049)
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 M. Van Veldhuizen, "Highorder methods for a singularly perturbed problem," Numer. Math., v. 30, 1978, pp. 267279. MR 0501937 (58:19156)
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 R. Weiss, "An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems," Math. Comp., v. 42, 1984, pp. 4167. MR 725984 (86b:65085)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708786900
PII:
S 00255718(1987)08786900
Article copyright:
© Copyright 1987
American Mathematical Society
