Geometric integrators for piecewise smooth Hamiltonian systems
ESAIM: Mathematical Modelling and Numerical Analysis (2008)
- Volume: 42, Issue: 2, page 223-241
- ISSN: 0764-583X
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topChartier, 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
top- R.J. Di Perna and P.L. Lions, 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).
- E. Hairer, Important aspects of geometric numerical integration. J. Sci. Comput.25 (2005) 67–81.
- 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).
- M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations. Numer. Math.83 (1999) 403–426.
- A. Kvaerno and B. Leimkuhler, A time-reversible, regularized, switching integrator for the n-body problem. SIAM J. Sci. Comput.22 (2000) 1016–1035.
- B. Laird and B. Leimkuhler, A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys.98 (2000) 309–316.
- 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.
- R.I. McLachlan and G.R.W. Quispel, Geometric integration of conservative polynomial ODEs. Appl. Numer. Math.45 (2003) 411–418.
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