# An arbitrary order time-stepping algorithm for tracking particles in inhomogeneous magnetic fields

## TL;DR

This paper introduces an improved high-order Boris-SDC algorithm for particle tracking in magnetic fields, enhancing accuracy and stability while reducing computational costs compared to the standard Boris method.

## Contribution

The authors develop a new Boris-SDC algorithm using GMRES-based convergence acceleration, achieving arbitrary order accuracy with better energy stability and efficiency.

## Key findings

- Better accuracy at lower computational cost
- Enhanced long-term energy stability
- Effective in magnetic mirror and equilibrium scenarios

## Abstract

The Lorentz equations describe the motion of electrically charged particles in electric and magnetic fields and are used widely in plasma physics. The most popular numerical algorithm for solving them is the Boris method, a variant of the St\"ormer-Verlet algorithm. Boris' method is phase space volume conserving and simulated particles typically remain near the correct trajectory. However, it is only second order accurate. Therefore, in scenarios where it is not enough to know that a particle stays on the right trajectory but one needs to know where on the trajectory the particle is at a given time, Boris method requires very small time steps to deliver accurate phase information, making it computationally expensive. We derive an improved version of the high-order Boris spectral deferred correction algorithm (Boris-SDC) by adopting a convergence acceleration strategy for second order problems based on the Generalised Minimum Residual (GMRES) method. Our new algorithm is easy to implement as it still relies on the standard Boris method. Like Boris-SDC it can deliver arbitrary order of accuracy through simple changes of runtime parameter but possesses better long-term energy stability. We demonstrate for two examples, a magnetic mirror trap and the Solev'ev equilibrium, that the new method can deliver better accuracy at lower computational cost compared to the standard Boris method. While our examples are motivated by tracking ions in the magnetic field of a nuclear fusion reactor, the introduced algorithm can potentially deliver similar improvements in efficiency for other applications.