In this paper, we consider 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 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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {223--241}, publisher = {EDP-Sciences}, volume = {42}, number = {2}, year = {2008}, doi = {10.1051/m2an:2008006}, mrnumber = {2405146}, zbl = {1145.65110}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008006/} }
TY - JOUR AU - Chartier, Philippe AU - Faou, Erwan TI - Geometric integrators for piecewise smooth hamiltonian systems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 223 EP - 241 VL - 42 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008006/ DO - 10.1051/m2an:2008006 LA - en ID - M2AN_2008__42_2_223_0 ER -
%0 Journal Article %A Chartier, Philippe %A Faou, Erwan %T Geometric integrators for piecewise smooth hamiltonian systems %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 223-241 %V 42 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008006/ %R 10.1051/m2an:2008006 %G en %F M2AN_2008__42_2_223_0
Chartier, Philippe; Faou, Erwan. Geometric integrators for piecewise smooth hamiltonian systems. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 2, pp. 223-241. doi : 10.1051/m2an:2008006. http://www.numdam.org/articles/10.1051/m2an:2008006/
[1] Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 3 (1995) 511-547. | MR | Zbl
and ,[2] Measure theory and fine properties of functions. CRC Press (1992). | MR | Zbl
and ,[3] Important aspects of geometric numerical integration. J. Sci. Comput. 25 (2005) 67-81. | MR
,[4] Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer, Berlin (2002). | MR | Zbl
, and ,[5] A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83 (1999) 403-426. | MR | Zbl
and ,[6] A time-reversible, regularized, switching integrator for the n-body problem. SIAM J. Sci. Comput. 22 (2000) 1016-1035. | MR | Zbl
and ,[7] A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys. 98 (2000) 309-316.
and ,[8] Renormalized solutions of some transport equations with partially velocities and applications. Ann. Mat. Pura Appl. 1 (2004) 97-130. | MR
and ,[9] Geometric integration of conservative polynomial ODEs. Appl. Numer. Math. 45 (2003) 411-418. | MR | Zbl
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