Abstract
At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff’ situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff’) case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is present.
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References
Benettin, G., Galgani, L., and Giorgilli, A. (1987). 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, 87–103.
Benettin, G., and Giorgilli, A. (1994). On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms.J. Statist. Phys. 74, 1117–1143.
Cohen, D., Hairer, E., and Lubich, C. (2003). Modulated Fourier expansions of highly oscillatory differential equations.Found. Comput. Math. 3, 327–345.
Cohen, D., Hairer, E., and Lubich, C. (2004). Numerical energy conservation for multifrequency oscillatory differential equations. To appear inBIT.
García-Archilla, B., Sanz-Serna, J. M., and Skeel, R. D. (1999). Long-time-step methods for oscillatory differential equations.SIAM J. Sci. Comput. 20, 930–963.
Gautschi, W. (1961). Numerical integration of ordinary differential equations based on trigonometric polynomials.Numer. Math. 3, 381–397.
Hairer, E., and Hairer, M. (2003). GniCodes — Matlab programs for geometric numerical integration.In: Frontiers in Numerical Analysis (Durham 2002), Springer, Berlin.
Hairer, E., and Lubich, C. (2000). Long-time energy conservation of numerical methods for oscillatory differential equations.SIAM J. Numer. Anal. 38, 414–441.
Hairer, E., Lubich, C., and Wanner, G. (2002).Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics 31. Springer, Berlin.
Hairer, E., Lubich, C., and Wanner, G. (2003). Geometric numerical integration illustrated by the Störmer-Verlet method.Acta Numerica 12, 399–450.
Hairer, E., Nørsett, S. P., and Wanner, G. (1993).Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Series in Computational Mathematics 8, 2nd ed., Springer, Berlin.
Hochbruck, M., and Lubich, C. (1999). A Gautschi-type method for oscillatory second-order differential equations.Numer. Math. 83, 403–426.
Murua, A. (1994).Métodos simplécticos desarrollables en P-series. PhD thesis, Univ. Valladolid.
Reich, S. (1999). Backward error analysis for numerical integrators.SIAM J. Numer. Anal. 36, 1549–1570.
Tang, Y.-F. (1994). Formal energy of a symplectic scheme for Hamiltonian systems and its applications I.Comput. Math. Applic. 27, 31–39.
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Hairer, E. Important aspects of geometric numerical integration. J Sci Comput 25, 67–81 (2005). https://doi.org/10.1007/BF02728983
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DOI: https://doi.org/10.1007/BF02728983