Summary
Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail.
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References
G. Benettin, L. Galgani & 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) 87–103.
M. Born, V. Fock, Beweis des Adiabatensatzes, Zs. Physik 51 (1928) 165–180.
F. Bornemann, Homogenization in Time of Singularly Perturbed Mechanical Systems, Springer LNM 1687 (1998).
F. A. Bornemann, P. Nettesheim, B. Schmidt, & C. Schütte, An explicit and symplectic integrator for quantum-classical molecular dynamics, Chem. Phys. Lett., 256 (1996) 581–588.
F. A. Bornemann, C. Schütte, On the singular limit of the quantum-classical molecular dynamics model, SIAM J. Appl. Math., 59 (1999) 1208–1224.
D. Cohen, Analysis and numerical treatment of highly oscillatory differential equations, Doctoral Thesis, Univ. de Genève (2004).
D. Cohen, Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems, IMA J. Numer. Anal. 26 (2006) 34–59.
D. Cohen, E. Hairer & C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations, Found. Comput. Math. 3 (2003) 327–345.
D. Cohen, E. Hairer & C. Lubich, Numerical energy conservation for multifrequency oscillatory differential equations, BIT 45 (2005) 287–305.
I. Degani & J. Schiff, RCMS: Right correction Magnus series approach for integration of linear ordinary differential equations with highly oscillatory solution, Report, Weizmann Inst. Science, Rehovot, 2003.
P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions, Z. angew. Math. Phys. 30 (1979) 177–189.
W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci. 1 (2003) 423–436.
B. Engquist & Y. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comp. 74 (2005) 1707–1742.
E. Faou & C. Lubich, A Poisson integrator for Gaussian wavepacket dynamics, Report, 2004. To appear in Comp. Vis. Sci.
B. García-Archilla, J. Sanz-Serna, R. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput. 20 (1999) 930–963.
W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961) 381–397.
V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math. 100 (2005) 71–89.
V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math. 102 (2005) 61–66.
V. Grimm & M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A 39 (2006)
H. Grubmüller, H. Heller, A. Windemuth & K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Sim. 6 (1991) 121–142.
E. Hairer, C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Num. Anal. 38 (2000) 414–441.
E. Hairer, C. Lubich & G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics 31. 2nd ed., 2006.
E. Hairer, C. Lubich & G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numerica (2003) 399–450.
J. Henrard, The adiabatic invariant in classical mechanics, Dynamics reported, New series. Vol. 2, Springer, Berlin (1993) 117–235.
J. Hersch, Contributionà la méthode aux différences, Z. angew. Math. Phys. 9a (1958) 129–180.
M. Hochbruck & C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math. 83 (1999) 403–426.
M. Hochbruck & C. Lubich, A bunch of time integrators for quantum/ classical molecular dynamics, in P. Deuflhard et al. (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, Springer, Berlin 1999, 421–432.
M. Hochbruck & C. Lubich, Exponential integrators for quantum-classical molecular dynamics, BIT 39 (1999) 620–645.
M. Hochbruck & C. Lubich, On Magnus integrators for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 41 (2003) 945–963.
A. Iserles, On the global error of discretization methods for highly-oscillatory ordinary differential equations, BIT 42 (2002) 561–599.
A. Iserles, On the method of Neumann series for highly oscillatory equations, BIT 44 (2004) 473–488.
T. Jahnke, Numerische Verfahren für fast adiabatische Quantendynamik, Doctoral Thesis, Univ. Tübingen (2003).
T. Jahnke, Long-time-step integrators for almost-adiabatic quantum dynamics, SIAM J. Sci. Comput. 25 (2004) 2145–2164.
T. Jahnke & C. Lubich, Numerical integrators for quantum dynamics close to the adiabatic limit, Numer. Math. 94 (2003) 289–314.
C. Lasser & S. Teufel, Propagation through conical crossings: an asymptotic semigroup, Comm. Pure Appl. Math. 58 (2005) 1188–1230.
B. Leimkuhler & S. Reich, A reversible averaging integrator for multiple time-scale dynamics, J. Comput. Phys. 171 (2001) 95–114.
B. Leimkuhler & S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics 14, Cambridge University Press, Cambridge, 2004.
K. Lorenz, Adiabatische Integratoren für hochoszillatorische mechanische Systeme, Doctoral thesis, Univ. Tübingen (2006).
K. Lorenz, T. Jahnke & C. Lubich, Adiabatic integrators for highly oscillatory second order linear differential equations with time-varying eigendecomposition, BIT 45 (2005) 91–115.
P. Nettesheim, Mixed quantum-classical dynamics: a unified approach to mathematical modeling and numerical simulation, Thesis FU Berlin (2000).
P. Nettesheim & S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in P. Deuflhard et al. (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, Springer, Berlin (1999) 412–420.
P. Nettesheim & C. Schütte, Numerical integrators for quantum-classical molecular dynamics, in P. Deuflhard et al. (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, Springer, Berlin (1999) 412–420.
L. R. Petzold, L. O. Jay & J. Yen, Numerical solution of highly oscillatory ordinary differential equations, Acta Numerica 7 (1997) 437–483.
S. Reich, Multiple time scales in classical and quantum-classical molecular dynamics, J. Comput. Phys. 151 (1999) 49–73.
H. Rubin & P. Ungar, Motion under a strong constraining force, Comm. Pure Appl. Math. 10 (1957) 65–87.
F. Takens, Motion under the influence of a strong constraining force, Global theory of dynamical systems, Proc. Int. Conf., Evanston/Ill. 1979, Springer LNM 819 (1980) 425–445.
M. Tuckerman, B.J. Berne & G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys. 97 (1992) 1990–2001.
C. Zener, Non-adiabatic crossing of energy levels, Proc. Royal Soc. London, Ser. A 137 (1932) 696–702.
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Cohen, D., Jahnke, T., Lorenz, K., Lubich, C. (2006). Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review. In: Mielke, A. (eds) Analysis, Modeling and Simulation of Multiscale Problems. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35657-6_20
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DOI: https://doi.org/10.1007/3-540-35657-6_20
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