Summary.
This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions.
Similar content being viewed by others
References
Ascher, U.M., Reich, S.: On some difficulties in integrating highly oscillatory Hamiltonian systems. In: Proc. Computational Molecular Dynamics, Springer Lecture Notes, 1999, pp. 281–296
Cohen, D., Hairer, E., Lubich, Ch.: Modulated fourier expansions of highly oscillatory differential equations. Foundations of Comput. Maths. 3, 327–450 (2003)
García-Archilla, B., Sanz-Serna, J., Skeel, R.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 30(3), 930–963 (1998)
Grimm, V.: Exponentielle Integratoren als Lange-Zeitschritt-Verfahren für oszillatorische Differentialgleichungen zweiter Ordnung, PhD thesis, Mathematisches Institut, Universität Düsseldorf, Germany, 2002
Grubmüller, H.: Dynamiksimulation sehr großer Makromoleküle auf einem Parallelrechner, PhD thesis, Physik-Dept. der Tech. Univ. München, 1994
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)
Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration. Springer-Verlag, 2002
Hochbruck, M., Lubich, Ch.: A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics. In: P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, volume 4 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 1997, pp. 421–432
Hochbruck, M., Lubich, Ch.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)
Hochbruck, M., Lubich, Ch.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)
Hochbruck, M., Lubich, Ch., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. 19, 1552–1574 (1998)
Iserles, A.: Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appld. Num. Anal. 43, 145–160 (2002)
Izaguirre, J.A., Reich, S., Skeel, R.D.: Longer time steps for molecular dynamics. J. Chemical Phys. 110(20), 9853–9864 (1999)
Petzold, L., Jay, L., Yen, J.: Numerical Solution of Highly Oscillatory Ordinary Differential Equations. Acta Numerica 6, 437–484 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 65L05, 65L70
Acknowledgement I am grateful to Marlis Hochbruck and Christian Lubich for helpful discussions on the subject.
Rights and permissions
About this article
Cite this article
Grimm, V. On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations. Numer. Math. 100, 71–89 (2005). https://doi.org/10.1007/s00211-005-0583-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0583-8