Geometric integrators for piecewise smooth hamiltonian systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, p. 223-241

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.

DOI : https://doi.org/10.1051/m2an:2008006
Classification:  65L05,  65L06,  65L20
Keywords: hamiltonian systems, symplecticity, volume-preservation, energy-preservation, B-splines, weak order
@article{M2AN_2008__42_2_223_0,
     author = {Chartier, Philippe and Faou, Erwan},
     title = {Geometric integrators for piecewise smooth hamiltonian systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     pages = {223-241},
     doi = {10.1051/m2an:2008006},
     zbl = {1145.65110},
     mrnumber = {2405146},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_2_223_0}
}
Chartier, Philippe; Faou, Erwan. Geometric integrators for piecewise smooth hamiltonian systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, pp. 223-241. doi : 10.1051/m2an:2008006. http://www.numdam.org/item/M2AN_2008__42_2_223_0/

[1] R.J. Di Perna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 3 (1995) 511-547. | MR 1022305 | Zbl 0696.34049

[2] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992). | MR 1158660 | Zbl 0804.28001

[3] E. Hairer, Important aspects of geometric numerical integration. J. Sci. Comput. 25 (2005) 67-81. | MR 2231943

[4] 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). | MR 1904823 | Zbl 0994.65135

[5] M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83 (1999) 403-426. | MR 1715573 | Zbl 0937.65077

[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. | MR 1785344 | Zbl 0993.70003

[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 w 1,1 velocities and applications. Ann. Mat. Pura Appl. 1 (2004) 97-130. | MR 2044334 | Zbl pre05058532

[9] R.I. Mclachlan and G.R.W. Quispel, Geometric integration of conservative polynomial ODEs. Appl. Numer. Math. 45 (2003) 411-418. | MR 1983487 | Zbl 1021.65064