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

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 Störmer-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.

Performance of the BGSDC integrator for computing fast ion trajectories in nuclear fusion reactors

“…see Tretiak et al for a discussion [18]. A geometric trick was introduced by Boris in 1970 [23] to avoid the seemingly implicit dependence on v n+1 .…”

“…where Q ∆t has a block structure with each block being a lower triangular matrix [18]. To apply GMRES to the preconditioned problem, a second function is required that can solve…”

“…using Boris' trick to compute the velocities [18]. Note that for a single particle, where F = f , with given initial values x n , v n at the beginning of the time step, setting b (11) 12) and ( 13) correspond to a total of three Boris steps (3) with step sizes ∆τ 1 , ∆τ 2 and ∆τ 3 .…”

“…For nonlinear collocation problems, it is possible to apply a Newton iteration and use GMRES-SDC to solve the linear inner problems. For the case where the nonlinearity is due to the magnetic field depending on x, this was found not to be competitive [18]. Instead, we linearize the collocation problem by freezing the magnetic field after the first sweep provides values x 0 m for m = 1, .…”

“…Tretiak and Ruprecht [18] introduce BGSDC, a new high-order algorithm for solving the Lorentz equations based on a combination of spectral deferred corrections, a Generalized Minimal Residual (GMRES) iteration and the Boris integrator. They show improvements in computational performance over Boris for individual particle trajectories in a mirror trap as well as trapped and passing particles in a Solev'ev equilibrium.…”