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Superconvergence for multistep collocation

Authors: Ivar Lie and Syvert P. Nørsett
Journal: Math. Comp. 52 (1989), 65-79
MSC: Primary 65L05
MathSciNet review: 971403
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Abstract: One-step collocation methods are known to be a subclass of implicit Runge-Kutta methods. Further, one-leg methods are special multistep one-point 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 $ m + k - 1$. However, by choosing the collocation points in the right way, order $ 2m + k - 1$ is obtained as the maximum. There are $ \left( {\begin{array}{*{20}{c}} {m + k - 1} \\ {k - 1} \\ \end{array} } \right)$ sets of such "multistep Gaussian" collocation points.

References [Enhancements On Off] (What's this?)

  • [1] 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 Runge-Kutta methods." Manuscript, 1985.
  • [3] K. Burrage & P. Moss, "Simplifying assumptions for the order of partitioned multivalue methods," BIT, v. 20, 1980, pp. 452-465. MR 605903 (83k:65055)
  • [4] G. Dahlquist, Some Properties of Linear Multistep Methods and One-Leg Methods for Ordinary Differential Equations, Report TRITA-NA-7904, KTH, Stockholm, 1979.
  • [5] G. Dahlquist, "On one-leg multistep methods," SIAM J. Numer. Anal., v. 20, 1983, pp. 1130-1138. MR 723829 (85h:65144)
  • [6] A. Guillon & F. L. Soulé, "La résolution numérique des problèmes differentiels aux conditions initiales par des méthodes de collocation," RAIRO Anal. Numér. Ser. Rouge, v. R-3 1969, pp. 17-44. MR 0280008 (43:5729)
  • [7] V. I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, 1962. MR 0144464 (26:2008)
  • [8] I. Lie, k-Step 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. Munthe-Kaas, On the Number of Gaussian Points for Multistep Collocation, Technical report, University of Trondheim, Dept. of Numerical Mathematics, 1986.
  • [11] S. P. Nørsett, "Runge Kutta methods with a multiple eigenvalue only," BIT, v. 16, 1976, pp. 388-393. MR 0440928 (55:13796)
  • [12] S. P. Nørsett, Collocation and Perturbed Collocation Methods, Lecture Notes in Math., vol. 773 (G. A. Watson, ed.), Springer-Verlag, Berlin and New York, 1980. MR 569466 (81h:65078)
  • [13] S. P. Nørsett &. G. Wanner, "The real-pole sandwich for rational approximations and oscillation equations," BIT, v. 19, 1979, pp. 79-94. MR 530118 (81d:65040)
  • [14] S. P. Nørsett & G. Wanner, "Perturbed collocation and Runge-Kutta methods," Numer. Math., v. 38, 1981, pp. 193-208. 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, "One-step collocation: Uniform superconvergence, predictor-corrector methods, local error estimates," SIAM J. Numer. Anal., v. 22, 1985, pp. 1135-1152. MR 811188 (86m:65084)

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Article copyright: © Copyright 1989 American Mathematical Society

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