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Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field

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Abstract

The differential equations of motion of a charged particle in a strong non-uniform magnetic field have the magnetic moment as an adiabatic invariant. This quantity is nearly conserved over long time scales covering arbitrary negative powers of the small parameter, which is inversely proportional to the strength of the magnetic field. The numerical discretisation is studied for a variational integrator that is an analogue for charged-particle dynamics of the Störmer–Verlet method. This numerical integrator is shown to yield near-conservation of a modified magnetic moment and a modified energy over similarly long times. The proofs for both the continuous and the discretised equations use modulated Fourier expansions with state-dependent frequencies and eigenvectors.

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Notes

  1. We use the notation \({{\,\mathrm{sinc}\,}}(\xi ) = \sin (\xi )/\xi \).

  2. Actually, for \(j=0\) the sum is over \(0\le m\le N+2\). However, the term with \(m= N+2\) is of size \({{\mathcal {O}}}(\varepsilon ^{N+2})\) and can be absorbed in the remainder.

  3. We thank Ludwig Gauckler for drawing our attention to this idea in connection with the problem considered in [8].

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Acknowledgements

We thank Josh Burby for adverting us to the unpublished report [13] by Kruskal from 1958, and we thank Sebastian Reich for his interest in this work. The research for this article has been partially supported by the Fonds National Suisse, Project No. 200020_159856, and by Deutsche Forschungsgemeinschaft, SFB 1173.

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  1. Dept. de Mathématiques, Univ. de Genève, 1211, Geneva 24, Switzerland

    Ernst Hairer

  2. Mathematisches Institut, Univ. Tübingen, 72076, Tübingen, Germany

    Christian Lubich

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  1. Ernst Hairer
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  2. Christian Lubich
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Correspondence to Christian Lubich.

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Hairer, E., Lubich, C. Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field. Numer. Math. 144, 699–728 (2020). https://doi.org/10.1007/s00211-019-01093-z

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  • DOI: https://doi.org/10.1007/s00211-019-01093-z

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