Symmetric multistep methods for charged-particle dynamics
The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 205-218.

A class of explicit symmetric multistep methods is proposed for integrating the equations of motion of charged particles in an electro-magnetic 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.

Published online:
Classification: 65L06,  65P10,  78A35,  78M25
Keywords: linear multistep method, charged particle, magnetic field, energy conservation, backward error analysis, modified differential equation, modulated Fourier expansion
     author = {Ernst Hairer and Christian Lubich},
     title = {Symmetric multistep methods for charged-particle dynamics},
     journal = {The SMAI journal of computational mathematics},
     pages = {205--218},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {3},
     year = {2017},
     doi = {10.5802/smai-jcm.25},
     mrnumber = {3716756},
     zbl = {1416.78037},
     language = {en},
     url = {}
Ernst Hairer; Christian Lubich. Symmetric multistep methods for charged-particle dynamics. The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 205-218. doi : 10.5802/smai-jcm.25.

[1] J. P. Boris Relativistic plasma simulation-optimization of a hybrid code, Proceeding of Fourth Conference on Numerical Simulations of Plasmas (1970), pp. 3-67

[2] P. Console; E. Hairer; C. Lubich Symmetric multistep methods for constrained Hamiltonian systems, Numerische Mathematik, Volume 124 (2013), pp. 517-539 | Article | MR 3066038 | Zbl 1290.65125

[3] G. Dahlquist Stability and error bounds in the numerical integration of ordinary differential equations, Trans. of the Royal Inst. of Techn., Stockholm, Sweden, Volume 130 (1959) | MR 102921 | Zbl 0085.33401

[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., Volume 301 (2015), pp. 489-493 | Article | MR 3402744 | Zbl 1349.37083

[5] E. Hairer; C. Lubich Symmetric multistep methods over long times, Numer. Math., Volume 97 (2004), pp. 699-723 | Article | MR 2127929 | Zbl 1060.65074

[6] E. Hairer; C. Lubich Energy behaviour of the Boris method for charged-particle dynamics, Submitted for publication (2017) | Zbl 1404.65309

[7] E. Hairer; C. Lubich; G. Wanner Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, 2006 | Zbl 1094.65125

[8] E. Hairer; S. P. Nørsett; G. Wanner Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8, Springer, Berlin, 1993 | Zbl 0789.65048

[9] Y. He; Z. Zhou; Y. Sun; J. Liu; H. Qin Explicit K-symplectic algorithms for charged particle dynamics, Phys. Lett. A, Volume 381 (2017) no. 6, pp. 568-573 | Article | MR 3590622 | Zbl 1372.78010

[10] M. Tao Explicit high-order symplectic integrators for charged particles in general electromagnetic fields, J. Comput. Phys., Volume 327 (2016), pp. 245-251 | Article | MR 3564337 | Zbl 1373.78048

[11] S. D. Webb Symplectic integration of magnetic systems, J. Comput. Phys., Volume 270 (2014), pp. 570-576 | Article | MR 3209402 | Zbl 1349.37085

[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, Volume 94 (2016) no. 1, 013205 pages | Article