Abstract
In this article, conditions for the preservation of quadratic and Hamiltonian invariants by numerical methods which can be written as B-series are derived in a purely algebraical way. The existence of a modified invariant is also investigated and turns out to be equivalent, up to a conjugation, to the preservation of the exact invariant. A striking corollary is that a symplectic method is formally conjugate to a method that preserves the Hamitonian exactly. Another surprising consequence is that the underlying one-step method of a symmetric multistep scheme is formally conjugate to a symplectic P-series when applied to Newton’s equations of motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brouder Ch. (2004). Trees, renormalization and differential equations. BIT 44(3):425–438
Bochev P.B., Scovel C. (1994). On quadratic invariants and symplectic structure. BIT 34:337–345
Butcher, J.C.: The effective order of Runge-Kutta methods. In: Morris, J.Ll. (ed) Proceedings of conference on the numerical solution of differential equations, vol 109 of Lecture Notes in Math., pp 133–139 (1969)
Butcher J.C. (1972). An algebraic theory of integration methods. Math. Comput. 26:79–106
Butcher J.C. (1987). The numerical analysis of ordinary differential equations. Runge–Kutta and general linear methods. Wiley, Chichester
Calvo M.P., Sanz-Serna J.M. (1994). Canonical B-series. Numer. Math. 67:161–175
Chartier P., Hairer E., Vilmart G. (2005). A substitution law for B-series vector fields. Technical Report 5498, INRIA, France
Connes A., Kreimer D. (1998). Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199: 203–242
Faou E., Hairer E., Pham T.L. (2004). Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44:699–709
Hairer E. (1994). Backward analysis of numerical integrators and symplectic methods. Ann. Numer. Math. 1:107–132
Hairer E. (1999). Backward error analysis for multistep methods. Numer. Math. 84:199–232
Hairer E., Lubich C. (2004). Symmetric multistep methods over long times. Numer. Math. 97:699–723
Hairer E., Wanner G. (1974). On the Butcher group and general multi-value methods. Computing 13:1–15
Hairer E., Nørsett S.P., Wanner G. (1993). Solving Ordinary differential equations. I Nonstiff problems Springer Series in Computational Mathematics, vol 8, 2nd edn. Springer, Berlin Heidelberg New york
Hairer E., Lubich C., Wanner G. (2002). Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Heidelberg New york
Kirchgraber U. (1986). Multi-step methods are essentially one-step methods. Numer. Math. 48:85–90
Merson, R.H.: An operational method for the study of integration processes. In: Proceedings of symposium data processing weapons research establishment, Salisbury Australia, pp 110–1 to 110–25 (1957)
Murua A. (1999). Formal series and numerical integrators, part i: systems of ODEs and symplectic integrators. Appl. Numer. Math. 29:221–251
Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Accepted for publication in FOCM, 2005
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chartier, P., Faou, E. & Murua, A. An Algebraic Approach to Invariant Preserving Integators: The Case of Quadratic and Hamiltonian Invariants. Numer. Math. 103, 575–590 (2006). https://doi.org/10.1007/s00211-006-0003-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-006-0003-8