BIT Numerical Mathematics

, Volume 45, Issue 1, pp 91–115

Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition


DOI: 10.1007/s10543-005-2637-9

Cite this article as:
Lorenz, K., Jahnke, T. & Lubich, C. Bit Numer Math (2005) 45: 91. doi:10.1007/s10543-005-2637-9


Numerical integrators for second-order differential equations with time-dependent high frequencies are proposed and analysed. We derive two such methods, called the adiabatic midpoint rule and the adiabatic Magnus method. The integrators are based on a transformation of the problem to adiabatic variables and an expansion technique for the oscillatory integrals. They can be used with far larger step sizes than those required by traditional schemes, as is illustrated by numerical experiments. We prove second-order error bounds with step sizes significantly larger than the almost-period of the fastest oscillations.


oscillatory problem numerical integrator long-time-step method multiple time scales adiabatic transformation 

Copyright information

© Springer 2005

Authors and Affiliations

  • Katina Lorenz
    • 1
  • Tobias Jahnke
    • 2
  • Christian Lubich
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Institut für Mathematik II, BioComputing GroupFreie Universität BerlinBerlinGermany

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