Abstract
The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the Störmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included.
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References
G. Benettin, L. Galgani, and A. Giorgilli, Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part I, Comm. Math. Phys., 113 (1987), pp. 87–103.
G. Benettin, L. Galgani, and A. Giorgilli, Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part II, Comm. Math. Phys., 121 (1989), pp. 557–601.
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74 (1994), pp. 1117–1143.
D. Cohen, E. Hairer, and C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations, Foundations of Comput. Math., 3 (2003), pp. 327–345.
P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions, Z. Angew. Math. Phys., 30 (1979), pp. 177-189.
B. García-Archilla, J. M. Sanz-Serna, and R. D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 20 (1999), pp. 930–963.
W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), pp. 381–397.
E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., 76 (1997), pp. 441–462. Erratum: http://www.unige.ch/math/folks/hairer/.
E. Hairer and C. Lubich, Energy conservation by Störmer-type numerical integrators, in D. F. Griffiths, G. A. Watson (eds), Numerical Analysis 1999, CRC Press LLC, 2000, pp. 169–190.
E. Hairer and C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM J. Numer. Anal., 38 (2000), pp. 414–441.
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.
M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), pp. 403–426.
S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), pp. 1549–1570.
S. Reich, Preservation of adiabatic invariants under symplectic discretization, Appl. Numer. Math., 29 (1999), pp. 45–55.
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AMS subject classification (2000)
65L05, 65P10
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Cohen, D., Hairer, E. & Lubich, C. Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations. Bit Numer Math 45, 287–305 (2005). https://doi.org/10.1007/s10543-005-7121-z
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DOI: https://doi.org/10.1007/s10543-005-7121-z