Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields

TL;DR

This paper introduces a quasi-geometric numerical method for guiding-center orbit integration in complex toroidal magnetic fields, offering high accuracy, conservation of invariants, and improved computational efficiency over traditional methods.

Contribution

The paper presents a novel quasi-geometric integration technique that exactly treats motion in piecewise linear fields, enhancing efficiency and invariant conservation in guiding-center orbit simulations.

Findings

Method preserves energy, magnetic moment, and phase space volume.

Achieves an order of magnitude higher efficiency than Runge-Kutta methods.

Demonstrates stable long-term orbit dynamics in complex magnetic configurations.

Abstract

A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly and thus preserves the non-canonical symplectic form. The algorithm itself is only quasi-geometric due to a series expansion in the orbit parameter. For practical purposes an expansion to the fourth order retains geometric properties down to computer accuracy in typical examples. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates stable long-term orbit dynamics conserving invariants. In Monte Carlo evaluation of transport coefficients, the computational efficiency of quasi-geometric integration is an order of magnitude higher than with a standard fourth order Runge-Kutta integrator.