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

## TL;DR

This paper develops and tests symplectic numerical methods for guiding-center orbits in magnetic confinement devices, improving long-term accuracy and computational efficiency over traditional schemes.

## Contribution

It introduces a generalized semi-implicit symplectic integration approach with coordinate transformations for plasma particle guiding-center motion.

## Key findings

- Symplectic methods outperform adaptive Runge-Kutta in long-term orbit simulations.
- The new approach achieves over three times speed-up in particle loss statistics.
- Coordinate transformations enable efficient parallel computations in complex magnetic fields.

## Abstract

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 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., Phys. Plasmas 21, 32504 (2014)]. Here a generalization of this approach with emphasis on semi-implicit partitioned schemes is described together with methods to enhance their 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.