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.
Highlights
The Lorentz equations x = v, (1a)v = α [E(x, t) + v × B(x, t))] =: f(x, v)(1b) model movement of charged particles in electro-magnetic fields
The more complex interplay between Generalised Minimum Residual (GMRES) and Picard iterations does not allow a simple heuristic like two orders per iteration that was found for non-accelerated Boris-Spectral deferred correction (SDC) [18], these results show that BGSDC can deliver high orders of convergence by changing the runtime parameter M and (Kgmres, Kpicard)
The paper introduces Boris-GMRES-SDC (BGSDC), a new high order algorithm to numerically solve the Lorentz equations based on the widely-used Boris method
Summary
(1b) model movement of charged particles in electro-magnetic fields. Here, x(t) is a vector containing all particle position at some time t, v(t) contains all particle velocities, α is the charge-to-mass ratio, E the electric field (both externally applied and internally generated from particle interaction) and B the magnetic field. One example are particle-wave interactions triggering Alfvén instabilities due to resonances between orbit frequencies and wave velocities [7] Because it is only second order accurate, the Boris method requires very small time steps, creating substantial computational cost. SDC provides dense output and allows to generate a high order solution anywhere within a time step We use this feature to accurately compute the turning points of particles in a magnetic mirror. Substantial differences between Verlet and Leapfrog seem only to arise in simulations with very large time steps with nearly no significant digits left (phase errors well above 10−1), where staggered Boris shows better stability In such regimes, BGSDC is not going to be competitive anyway so that we focus here on the simpler Verlet-based variant. See Birdsall and Langdon [23, Section 4–4] for the geometric derivation
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