Superconvergence for multistep collocation
Authors:
Ivar Lie and Syvert P. Nørsett
Journal:
Math. Comp. 52 (1989), 6579
MSC:
Primary 65L05
MathSciNet review:
971403
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Abstract: Onestep collocation methods are known to be a subclass of implicit RungeKutta methods. Further, oneleg methods are special multistep onepoint collocation methods. In this paper we extend both of these collocation ideas to multistep collocation methods with k previous meshpoints and m collocation points. By construction, the order is at least . However, by choosing the collocation points in the right way, order is obtained as the maximum. There are sets of such "multistep Gaussian" collocation points.
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 K. Burrage, The Order Properties of Implicit Multivalue Methods for Ordinary Differential Equations, Report 176/84, Dept. of Computer Science, University of Toronto, Toronto, Canada.
 [2]
 K. Burrage, "High order algebraically stable multistep RungeKutta methods." Manuscript, 1985.
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 [5]
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 V. I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, 1962. MR 0144464 (26:2008)
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 I. Lie, kStep Collocations with One Collocation Point and Derivative Data, FFI/NOTAT83/7109, NDRE, Kjeller, Norway, 1983.
 [9]
 I. Lie, Multistep Collocation for Stiff Systems, Ph.D. thesis, Norwegian Institute of Technology, Dept. of Numerical Mathematics, Trondheim, 1985.
 [10]
 H. MuntheKaas, On the Number of Gaussian Points for Multistep Collocation, Technical report, University of Trondheim, Dept. of Numerical Mathematics, 1986.
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 S. P. Nørsett, "Runge Kutta methods with a multiple eigenvalue only," BIT, v. 16, 1976, pp. 388393. MR 0440928 (55:13796)
 [12]
 S. P. Nørsett, Collocation and Perturbed Collocation Methods, Lecture Notes in Math., vol. 773 (G. A. Watson, ed.), SpringerVerlag, Berlin and New York, 1980. MR 569466 (81h:65078)
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 S. P. Nørsett &. G. Wanner, "The realpole sandwich for rational approximations and oscillation equations," BIT, v. 19, 1979, pp. 7994. MR 530118 (81d:65040)
 [14]
 S. P. Nørsett & G. Wanner, "Perturbed collocation and RungeKutta methods," Numer. Math., v. 38, 1981, pp. 193208. MR 638444 (82m:65065)
 [15]
 J. M. Ortega & W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
 [16]
 M. Zennaro, "Onestep collocation: Uniform superconvergence, predictorcorrector methods, local error estimates," SIAM J. Numer. Anal., v. 22, 1985, pp. 11351152. MR 811188 (86m:65084)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909714035
PII:
S 00255718(1989)09714035
Article copyright:
© Copyright 1989
American Mathematical Society
