Superconvergence for multistep collocation

Authors:
Ivar Lie and Syvert P. Nørsett

Journal:
Math. Comp. **52** (1989), 65-79

MSC:
Primary 65L05

MathSciNet review:
971403

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 . 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.

**[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]**Kevin Burrage and Pamela Moss,*Simplifying assumptions for the order of partitioned multivalue methods*, BIT**20**(1980), no. 4, 452–465. MR**605903**, 10.1007/BF01933639**[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]**Germund Dahlquist,*On one-leg multistep methods*, SIAM J. Numer. Anal.**20**(1983), no. 6, 1130–1138. MR**723829**, 10.1137/0720082**[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. Ser. R-3, 17–44 (French). MR**0280008****[7]**Vladimir Ivanovich Krylov,*Approximate calculation of integrals*, Translated by Arthur H. Stroud, The Macmillan Co., New York-London, 1962, 1962. MR**0144464****[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. MR**0440928****[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. MR**569466****[13]**S. P. Nørsett and G. Wanner,*The real-pole sandwich for rational approximations and oscillation equations*, BIT**19**(1979), no. 1, 79–94. MR**530118**, 10.1007/BF01931224**[14]**S. P. Nørsett and G. Wanner,*Perturbed collocation and Runge-Kutta methods*, Numer. Math.**38**(1981/82), no. 2, 193–208. MR**638444**, 10.1007/BF01397089**[15]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[16]**Marino Zennaro,*One-step collocation: uniform superconvergence, predictor-corrector method, local error estimate*, SIAM J. Numer. Anal.**22**(1985), no. 6, 1135–1152. MR**811188**, 10.1137/0722068

Retrieve articles in *Mathematics of Computation*
with MSC:
65L05

Retrieve articles in all journals with MSC: 65L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0971403-5

Article copyright:
© Copyright 1989
American Mathematical Society