Geometric integrators for piecewise smooth Hamiltonian systems

Philippe Chartier; Erwan Faou

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 2, page 223-241
  • ISSN: 0764-583X

Abstract

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In this paper, we consider C1,1 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.

How to cite

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Chartier, Philippe, and Faou, Erwan. "Geometric integrators for piecewise smooth Hamiltonian systems." ESAIM: Mathematical Modelling and Numerical Analysis 42.2 (2008): 223-241. <http://eudml.org/doc/250360>.

@article{Chartier2008,
abstract = { In this paper, we consider C1,1 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. },
author = {Chartier, Philippe, Faou, Erwan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hamiltonian systems; symplecticity; volume-preservation; energy-preservation; B-splines; weak order.; weak order; convergence; splitting method; geometric integrator},
language = {eng},
month = {3},
number = {2},
pages = {223-241},
publisher = {EDP Sciences},
title = {Geometric integrators for piecewise smooth Hamiltonian systems},
url = {http://eudml.org/doc/250360},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Chartier, Philippe
AU - Faou, Erwan
TI - Geometric integrators for piecewise smooth Hamiltonian systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/3//
PB - EDP Sciences
VL - 42
IS - 2
SP - 223
EP - 241
AB - In this paper, we consider C1,1 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.
LA - eng
KW - Hamiltonian systems; symplecticity; volume-preservation; energy-preservation; B-splines; weak order.; weak order; convergence; splitting method; geometric integrator
UR - http://eudml.org/doc/250360
ER -

References

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  1. R.J. Di Perna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math.3 (1995) 511–547.  
  2. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992).  Zbl0804.28001
  3. E. Hairer, Important aspects of geometric numerical integration. J. Sci. Comput.25 (2005) 67–81.  Zbl1203.65115
  4. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics31. Springer, Berlin (2002).  Zbl0994.65135
  5. M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations. Numer. Math.83 (1999) 403–426.  Zbl0937.65077
  6. A. Kvaerno and B. Leimkuhler, A time-reversible, regularized, switching integrator for the n-body problem. SIAM J. Sci. Comput.22 (2000) 1016–1035.  Zbl0993.70003
  7. B. Laird and B. Leimkuhler, A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys.98 (2000) 309–316.  
  8. C. Le Bris and P.L. Lions, Renormalized solutions of some transport equations with partially w1,1 velocities and applications. Ann. Mat. Pura Appl.1 (2004) 97–130.  Zbl1170.35364
  9. R.I. McLachlan and G.R.W. Quispel, Geometric integration of conservative polynomial ODEs. Appl. Numer. Math.45 (2003) 411–418.  Zbl1021.65064

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