A Collocated Boris Integrator in Flux Coordinates: Balancing Accuracy, Conservation, Cost and Robustness

When the guiding-center description fails and the full gyromotion must be resolved for energetic particles in complex configurations like stellarators, charged-particle integrators must be formulated directly in the curvilinear flux coordinates.

The Boris algorithm, which adopts a staggered scheme in Cartesian coordinates, is phase-space-volume-preserving and second-order accurate; but a direct port to flux coordinates degrades the position update to first order, because the evolving basis vectors of the curvilinear frame make the starting-point metric deviate from the ideal midpoint metric.

We construct a collocated, midpoint-predicted Boris algorithm in flux coordinates, restoring second-order accuracy at the cost of one additional field evaluation per step.

In reactor-scale stellarator magnetic fields, the scheme recovers second-order convergence in every coordinate component, retains near-machine-precision energy conservation and a bounded magnetic moment, and demonstrates greater orbit robustness than Staggered Boris and RK4 at coarse time steps.