Large-stepsize integrators for charged-particle dynamics over multiple time scales

Abstract

The Boris algorithm, a closely related variational integrator and a newly proposed filtered variational integrator are studied when they are used to numerically integrate the equations of motion of a charged particle in a mildly non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For the Boris algorithm and the standard (unfiltered) variational integrator, satisfactory behaviour is only obtained when the component of the initial velocity orthogonal to the magnetic field is filtered out. The particle motion shows varying behaviour over multiple time scales: fast gyrorotation, guiding centre motion, slow perpendicular drift, near-conservation of the magnetic moment over very long times and conservation of energy for all times. Using modulated Fourier expansions of the exact and numerical solutions, it is analysed to which extent this behaviour is reproduced by the three numerical integrators used with large step sizes that do not resolve the fast gyrorotations.

Continuous-stage symplectic adapted exponential methods for charged-particle dynamics with arbitrary electromagnetic fields

“…If one considers the methods mentioned above for CPD with 0 < ǫ ≪ 1, the accuracy usually depends on 1/ǫ j for some j > 0 and this makes the methods inefficient for small ǫ. Concerning the methods for CPD with strong regime 0 < ǫ ≪ 1, to the best of my knowledge, most existed algorithms are devoted to the accuracy or long time near conservation [11,12,14,15,16,19,24,25,47]. Structure-preserving methods have not been considered and analysed for CPD with strong regime 0 < ǫ ≪ 1.…”

Asymptotically Preserving Particle Methods for Strongly Magnetized Plasmas in a Torus

“…Other approaches are based on the construction of efficient particle solvers for the original dynamics, to be used as a piece of a PIC scheme. Over the last decade, considerable efforts have been devoted to the design of such solvers and we refer the reader to [2,23,8,26,9,10,13,5,6,22,14,24,12,25,15,7] for both significant contributions and relevant entering gates to the now abundant literature. Along these years, roughly speaking, two kind of goals were assigned to the built numerical schemes.…”

“…For this reason, many of the structure preserving schemes built with classical tools from geometric numerical integration, such as multi-step schemes [13], variational schemes [26,14,24], or splitting schemes [25], fail to capture accurately the correct asymptotic behavior, and, even worse, many of them are only known to provide the desired structure preservation under upper size constraints on ∆t/ε, ∆t denoting the numerical time step. Unfortunately, so far proposed fixes for these geometric schemes, such as the introduction of ε-dependent filters [15], are consistent with the exact dynamics, as ∆t goes to zero, only under lower size restrictions on ∆t/ √ ε so that for the moment none of these provide a satisfactory behavior for the whole range of relevant physical and numerical parameters.…”