Summary
We generalize a result of Kirchgraber (1986) on multistep methods. We show that every strictly stable general linear method is essentially conjugate to a one step method of the same order. This result may be used to show that general properties of one step methods carry over to general linear methods. As examples we treat the existence of invariant curves and the construction of attracting sets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beyn, W.-J. (1987): On invariant closed curves for one-step methods. Numer. Math.51, 103–122
Brown, M., Hershenov, J. (1977): Periodic solutions of finite difference equtions. Quart. Appl. Math.35, 139–147
Butcher, J. (1987): The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley, New York
Cash, JR. (1983): Split linear multistep methods for the integration of stiff differential systems. Numer. Math.42, 299–310
Eirola, T. (1988): Invariant curves of one-step methods. BIT28, 113–122
Eirola, T., Nevanlinna, O. (1988): What do multistep methods approximate? Numer. Math.53, 559–569
Hirsch, M., Pugh, C., Shub, M. (1977): Invariant manifolds. Lect. Notes Math., No. 583. Springer, Berlin Heidelberg New York
Kirchgraber, U. (1986): Multistep methods are essentially one-step methods. Numer. Math.48, 85–90
Kloeden, P.E., Lorenz, J. (1986): Stable attracting sets in dynamical systems and in their one-step discretisations. SIAM J. Numer. Anal.23, 986–995
Kloeden, P.E., Lorenz, J. (1990): A note on multistep methods and attracting sets of dynamical systems. Numer. Math.56, 667–673
Nipp, K., Stoffer, D. (1992): Attractive in variant manifolds for maps: existence, smoothness and continuous dependence on the map. Research Report pp. 92–111, Applied Mathematics, ETH-Zürich
Shub, M. (1977): Global stability of dynamical systems. New York, Springer
Voss, D.A., Casper, M.J. (1989): Efficient split linear multistep methods for stiff ordinary differential equations. SIAM J. Sci. Stat. Comput.10, 990–999
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stoffer, D. General linear methods: connection to one step methods and invariant curves. Numer. Math. 64, 395–408 (1993). https://doi.org/10.1007/BF01388696
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01388696