Variational integrators for orbital problems using frequency estimation
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Variational integrators for orbital problems using frequency estimation Odysseas Kosmas1
· Sigrid Leyendecker2
Received: 22 June 2017 / Accepted: 14 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this work, we present a new derivation of higher order variational integration methods that exploit the phase lag properties for numerical integrations of systems with oscillatory solutions. More specifically, for the derivation of these integrators, the action integral along any curve segment is defined using a discrete Lagrangian that depends on the endpoints of the segment and on a number of intermediate points of interpolation. High order integrators are then obtained by writing down the discrete Lagrangian at any time interval as a weighted sum of the Lagrangians corresponding to a set of the chosen intermediate points. The respective positions and velocities are interpolated using trigonometric functions. The methods derived this way depend on a frequency, which in general needs to be accurately estimated. The new methods, which improve the phase lag characteristics by re-estimating the frequency at every time step, are presented and tested on the general N-body problem as numerical examples. Keywords Variational integrators · Discrete variational mechanics · Highly oscillatory problems · Phase lag analysis · General N-body problem Mathematics Subject Classification (2010) 49Mxx · 65Kxx
Communicated by: Tom Lyche Odysseas Kosmas
[email protected] 1
Modelling and Simulation Centre, MACE, University of Manchester, Sackville Street, Manchester, UK
2
Chair of Applied Dynamics, University of Erlangen-Nuremberg, Erlangen, 91058, Germany
O. Kosmas and S. Leyendecker
1 Introduction One of the most difficult problems in numerical solution of ordinary differential equations is the development of integrators for highly oscillatory systems, see [1]. Standard numerical schemes may require a huge number of time steps to track the oscillations, and even with small size steps they can alter the dynamics, unless the method is chosen adequately. In such cases, it is useful to adopt geometric integrators, that is, numerical schemes which preserve some of the geometric features of the dynamical system. These integrators can run in simulations for long time with less spurious effects (for instance, bad energy behavior for conservative systems) than the traditional ones, see [2–9]. So far, special techniques have been developed to improve the numerical integration of highly oscillatory problems, e.g. [1], and to derive methods as well as error bounds for families of quadrature methods that use finite difference approximations for the required derivatives. Alternatively, using the phase-lag property of [10], trigonometric fitting and phase-fitting techniques lead to methods based on variable coefficients which depend on the characteristic frequency of the problem, see [11]. The latter technique is known as exponential (or trigonometric) fitting and has been formulated
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