Abstract
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.
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AMS subject classification (2000)
65L05, 65L70.
Received February 2004. Accepted February 2005. Communicated by Syvert Nørsett.
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Lorenz, K., Jahnke, T. & Lubich, C. Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition. Bit Numer Math 45, 91–115 (2005). https://doi.org/10.1007/s10543-005-2637-9
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DOI: https://doi.org/10.1007/s10543-005-2637-9