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Superconvergence for multistep collocation. (English) Zbl 0658.65063
In a k-step collocation method for \(y'=f(x,y)\), \(y(x_ 0)=y_ 0\), the collocation polynomial for a subinterval \([x_ n,x_{n+1}]\) (given by two consecutive mesh points) is generated by also using previously computer approximations at the mesh points \(x_{n-k+1},...,x_ n\). This paper is concerned with the question of the attainable order p of k-step collocation methods using m distinct collocation points in each subinterval \([x_ n,x_{n+1}]:\) one has \(p=2m+k-1\), and it is shown that there exist \(\left( \begin{matrix} m+k-1\\ k-1\end{matrix} \right)\) different sets of “multistep Gaussian” collocation points for which this order is obtained. These sets are given explicit for \(m=2,3\), and \(k=1,2,3\). Forthcoming papers will deal with error estimation techniques and stability properties (for variable stepsize) of such multistep collocation methods.
Reviewer: H.Brunner

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
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[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. · Zbl 0637.65066
[2] K. Burrage, ”High order algebraically stable multistep Runge-Kutta methods.” Manuscript, 1985. · Zbl 0611.65046
[3] Kevin Burrage and Pamela Moss, Simplifying assumptions for the order of partitioned multivalue methods, BIT 20 (1980), no. 4, 452 – 465. · Zbl 0477.65051 · doi:10.1007/BF01933639 · doi.org
[4] G. Dahlquist, Some Properties of Linear Multistep Methods and One-Leg Methods for Ordinary Differential Equations, Report TRITA-NA-7904, KTH, Stockholm, 1979. · Zbl 0489.65044
[5] Germund Dahlquist, On one-leg multistep methods, SIAM J. Numer. Anal. 20 (1983), no. 6, 1130 – 1138. · Zbl 0529.65048 · doi:10.1137/0720082 · doi.org
[6] A. Guillou and J. L. Soulé, La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation, Rev. Française Informat. Recherche Opérationnelle 3 (1969), no. Sér. R-3, 17 – 44 (French). · Zbl 0214.15005
[7] Vladimir Ivanovich Krylov, Approximate calculation of integrals, Translated by Arthur H. Stroud, The Macmillan Co., New York-London, 1962, 1962.
[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] Syvert P. Nørsett, Runge-Kutta methods with a multiple real eigenvalue only, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 4, 388 – 393. · Zbl 0345.65036
[12] Syvert P. Nørsett, Collocation and perturbed collocation methods, Numerical analysis (Proc. 8th Biennial Conf., Univ. Dundee, Dundee, 1979), Lecture Notes in Math., vol. 773, Springer, Berlin, 1980, pp. 119 – 132. · Zbl 0447.65043
[13] S. P. Nørsett and G. Wanner, The real-pole sandwich for rational approximations and oscillation equations, BIT 19 (1979), no. 1, 79 – 94. · Zbl 0413.65011 · doi:10.1007/BF01931224 · doi.org
[14] S. P. Nørsett and G. Wanner, Perturbed collocation and Runge-Kutta methods, Numer. Math. 38 (1981/82), no. 2, 193 – 208. · Zbl 0471.65045 · doi:10.1007/BF01397089 · doi.org
[15] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. · Zbl 0241.65046
[16] Marino Zennaro, One-step collocation: uniform superconvergence, predictor-corrector method, local error estimate, SIAM J. Numer. Anal. 22 (1985), no. 6, 1135 – 1152. · Zbl 0608.65044 · doi:10.1137/0722068 · doi.org
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