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Geometric integrators for piecewise smooth Hamiltonian systems

Published online by Cambridge University Press:  27 March 2008

Philippe Chartier  and
Erwan Faou
Philippe Chartier
Affiliation:
IPSO, INRIA-Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France. [email protected]
Erwan Faou
Affiliation:
IPSO, INRIA-Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France. [email protected]
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Abstract

In this paper, we consider C 1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that it preserves the energy and the volume.

Keywords

Hamiltonian systemssymplecticityvolume-preservationenergy-preservationB-splinesweak order.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 2 , March 2008 , pp. 223 - 241
Copyright
© EDP Sciences, SMAI, 2008

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References

Di Perna, R.J. and Lions, P.L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 3 (1995) 511–547.
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992).
Hairer, E., Important aspects of geometric numerical integration. J. Sci. Comput. 25 (2005) 67–81. CrossRef
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer, Berlin (2002).
Hochbruck, M. and Lubich, C., Gautschi-type, A method for oscillatory second-order differential equations. Numer. Math. 83 (1999) 403–426. CrossRef
Kvaerno, A. and Leimkuhler, B., A time-reversible, regularized, switching integrator for the n-body problem. SIAM J. Sci. Comput. 22 (2000) 1016–1035. CrossRef
Laird, B. and Leimkuhler, B., A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys. 98 (2000) 309–316. CrossRef
Le Bris, C. and Lions, P.L., Renormalized solutions of some transport equations with partially w 1,1 velocities and applications. Ann. Mat. Pura Appl. 1 (2004) 97–130. CrossRef
McLachlan, R.I. and Quispel, G.R.W., Geometric integration of conservative polynomial ODEs. Appl. Numer. Math. 45 (2003) 411–418. CrossRef