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Ernst Hairer; Christian Lubich Symmetric multistep methods for chargedparticle dynamics SMAIJournal of computational mathematics, 3 (2017), p. 205218, doi: 10.5802/smaijcm.25 Article PDF Class. Math.: 65L06, 65P10, 78A35, 78M25 Keywords: linear multistep method, charged particle, magnetic field, energy conservation, backward error analysis, modified differential equation, modulated Fourier expansion Abstract A class of explicit symmetric multistep methods is proposed for integrating the equations of motion of charged particles in an electromagnetic field. The magnetic forces are built into these methods in a special way that respects the Lagrangian structure of the problem. It is shown that such methods approximately preserve energy and momentum over very long times, proportional to a high power of the inverse stepsize. We explain this behaviour by studying the modified differential equation of the methods and by analysing the remarkably stable propagation of parasitic solution components. Bibliography [2] P. Console, E. Hairer & C. Lubich, “Symmetric multistep methods for constrained Hamiltonian systems”, Numerische Mathematik 124 (2013), p. 517539 Article [3] G. Dahlquist, “Stability and error bounds in the numerical integration of ordinary differential equations”, Trans. of the Royal Inst. of Techn., Stockholm, Sweden 130 (1959) [4] C. L. Ellison, J. W. Burby & H. Qin, “Comment on “Symplectic integration of magnetic systems”: A proof that the Boris algorithm is not variational”, J. Comput. Phys. 301 (2015), p. 489493 Article [5] E. Hairer & C. Lubich, “Symmetric multistep methods over long times”, Numer. Math. 97 (2004), p. 699723 Article  MR 2127929 [6] E. Hairer & C. Lubich, “Energy behaviour of the Boris method for chargedparticle dynamics”, Submitted for publication (2017) [7] E. Hairer, C. Lubich & G. Wanner, Geometric Numerical Integration. StructurePreserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31, SpringerVerlag, Berlin, 2006 [8] E. Hairer, S. P. Nørsett & G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8, Springer, Berlin, 1993 [9] Y. He, Z. Zhou, Y. Sun, J. Liu & H. Qin, “Explicit $K$symplectic algorithms for charged particle dynamics”, Phys. Lett. A 381 (2017) no. 6, p. 568573 Article [10] M. Tao, “Explicit highorder symplectic integrators for charged particles in general electromagnetic fields”, J. Comput. Phys. 327 (2016), p. 245251 Article [11] S. D. Webb, “Symplectic integration of magnetic systems”, J. Comput. Phys. 270 (2014), p. 570576 Article [12] R. Zhang, H. Qin, Y. Tang, J. Liu, Y. He & J. Xiao, “Explicit symplectic algorithms based on generating functions for charged particle dynamics”, Physical Review E 94 (2016) no. 1 Article 
