Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices

Christopher G. Albert*, Sergei V. Kasilov, Winfried Kernbichler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The technique combines time-stepping in canonical coordinates with quadrature in non-canonical coordinates and is applicable in systems where a global transformation to canonical coordinates is known explicitly but its inverse is not. A fully implicit class of symplectic Runge-Kutta schemes has recently been introduced and applied to guiding-center motion by Zhang et al. (2014) [9]. Here a generalization of this approach with emphasis on semi-implicit partitioned schemes is described together with methods to enhance performance, in particular avoiding evaluation of non-canonical variables at full time steps. For application in toroidal plasma confinement configurations with nested magnetic flux surfaces a global canonicalization of coordinates for the guiding-center Lagrangian by a spatial transform is presented that allows for pre-computation of the required map in a parallel algorithm in the case of time-independent magnetic field geometry. Guiding-center orbits are studied in stationary magnetic equilibrium fields of an axisymmetric tokamak and a realistic three-dimensional stellarator configuration. Superior long-term properties of symplectic methods are demonstrated in comparison to a conventional adaptive Runge-Kutta scheme. Finally statistics of fast fusion alpha particle losses over their slowing-down time are computed in the stellarator field on a representative sample, reaching a speed-up of the symplectic Euler scheme by more than a factor three compared to usual Runge-Kutta schemes while keeping the same statistical accuracy and linear scaling with the number of computing threads.
Original languageEnglish
Article number109065
Number of pages23
JournalJournal of Computational Physics
Publication statusPublished - 15 Feb 2020


  • Guiding-center dynamics
  • Hamiltonian systems
  • Magnetic confinement
  • Numerical integration
  • Plasma
  • Symplectic integration

ASJC Scopus subject areas

  • Computational Mathematics
  • Physics and Astronomy(all)
  • Applied Mathematics
  • Numerical Analysis
  • Computer Science Applications
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)

Fields of Expertise

  • Information, Communication & Computing


  • NAWI Graz


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