Authors

Ernst Hairer, Christian Lubich

Abstract

The long-time behaviour of the Störmer–Verlet–leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved: the considered Hamiltonian systems have the action as an adiabatic invariant over long times that cover arbitrary negative powers of the small parameter. The Störmer–Verlet method approximately conserves a modified action and a modified total energy over a long time interval that covers a negative integer power of the small parameter. This power depends on the size of the product of the stepsize with the high frequency.

Keywords

65P10; 65L05; 34E13

Citation

  • Journal: Numerische Mathematik
  • Year: 2016
  • Volume: 134
  • Issue: 1
  • Pages: 119–138
  • Publisher: Springer Science and Business Media LLC
  • DOI: 10.1007/s00211-015-0766-x

BibTeX

@article{Hairer_2015,
  title={{Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency}},
  volume={134},
  ISSN={0945-3245},
  DOI={10.1007/s00211-015-0766-x},
  number={1},
  journal={Numerische Mathematik},
  publisher={Springer Science and Business Media LLC},
  author={Hairer, Ernst and Lubich, Christian},
  year={2015},
  pages={119--138}
}

Download the bib file

References

  • VI Arnold, Mathematical Aspects of Classical and Celestial Mechanics (1997)
  • Bornemann, F. Homogenization in Time of Singularly Perturbed Mechanical Systems. Lecture Notes in Mathematics (Springer Berlin Heidelberg, 1998). doi:10.1007/bfb0092091 – 10.1007/bfb0092091
  • Cohen, D., Gauckler, L., Hairer, E. & Lubich, C. Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions. BIT Numerical Mathematics vol. 55 705–732 (2014) – 10.1007/s10543-014-0527-8
  • Cohen, D., Hairer, E. & Lubich, Ch. Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations. BIT Numerical Mathematics vol. 45 287–305 (2005) – 10.1007/s10543-005-7121-z
  • Cohen, D., Hairer, E. & Lubich, C. Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations. Numerische Mathematik vol. 110 113–143 (2008) – 10.1007/s00211-008-0163-9
  • Cotter, C. J. & Reich, S. Adiabatic Invariance and Applications: From Molecular Dynamics to Numerical Weather Prediction. BIT Numerical Mathematics vol. 44 439–455 (2004) – 10.1023/b:bitn.0000046816.68632.49
  • FAOU, E., GAUCKLER, L. & LUBICH, C. PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION. Forum of Mathematics, Sigma vol. 2 (2014) – 10.1017/fms.2014.4
  • Gauckler, L. & Lubich, C. Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times. Foundations of Computational Mathematics vol. 10 275–302 (2010) – 10.1007/s10208-010-9063-3
  • Hairer, E. & Lubich, C. Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations. SIAM Journal on Numerical Analysis vol. 38 414–441 (2000) – 10.1137/s0036142999353594
  • E Hairer, Modulated, Expansions for Continuous and Discrete Oscillatory Systems, Foundations of Computational Mathematics, Budapest 2011, LMS Lecture Notes Series (2012)
  • E Hairer, Int. Math. Nachr. (2013)
  • Hairer, E., Lubich, C. & Wanner, G. Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica vol. 12 399–450 (2003) – 10.1017/s0962492902000144
  • E Hairer, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31 (2006)
  • Henrard, J. The Adiabatic Invariant in Classical Mechanics. Dynamics Reported 117–235 (1993) doi:10.1007/978-3-642-61232-9_4 – 10.1007/978-3-642-61232-9_4
  • McLachlan, R. I. & Stern, A. Modified Trigonometric Integrators. SIAM Journal on Numerical Analysis vol. 52 1378–1397 (2014) – 10.1137/130921118
  • Reich, S. Preservation of adiabatic invariants under symplectic discretization. Applied Numerical Mathematics vol. 29 45–55 (1999) – 10.1016/s0168-9274(98)00032-4
  • Reich, S. Smoothed Langevin dynamics of highly oscillatory systems. Physica D: Nonlinear Phenomena vol. 138 210–224 (2000) – 10.1016/s0167-2789(99)00200-6
  • Rubin, H. & Ungar, P. Motion under a strong constraining force. Communications on Pure and Applied Mathematics vol. 10 65–87 (1957) – 10.1002/cpa.3160100103
  • Stern, A. & Grinspun, E. Implicit-Explicit Variational Integration of Highly Oscillatory Problems. Multiscale Modeling & Simulation vol. 7 1779–1794 (2009) – 10.1137/080732936
  • Zhang, M. & Skeel, R. D. Cheap implicit symplectic integrators. Applied Numerical Mathematics vol. 25 297–302 (1997) – 10.1016/s0168-9274(97)00066-4