On the integration of equations of motion for particle-in-cell codes

Influence of oblique magnetic field on electron cross-field transport in a Hall effect thruster

“…22 Its main advantages include good stability, simplicity (only two evaluations of the fields per time step), and area-preservation. Considering a much lower velocity of ions and the fact that their trajectories are integrated with the same time step as those of electrons, the first order Euler method is sufficient to maintain a satisfactory accuracy.…”

Explicit time-reversible orbit integration in Particle In Cell codes with static homogeneous magnetic field

“…Eqs (2,6) readily show that the Boris integrator is time-reversible, even for non uniform magnetic fields. Indeed the Drift operator does not act on the particle velocity, and the Kick operator does not act on the position.…”

“…where the Boris Drift and Kick operators (D B and K B ) are defined in Eqs (2,3). If R ∆ϕ denotes a rotation of characteristic vector…”

“…In addition to not being time-reversible, Spreiter and Walter's algorithm has a Jacobian determinant slightly greater than unity [4], which usually leads to instability [2]. It is in fact possible to assert that the scheme is unconditionally unstable for any parameters except if Ω∆t = 0 and 0 ≤ ǫω 0 ∆t ≤ 2, in which case the algorithm is merely the standard unmagnetized leap-frog [1].…”

“…Other integrators such as the usual or midpoint second order Runge-Kutta [2] require two such evaluations per step, thus considerably increasing the computational cost. The second reason is that for stationary electric and magnetic fields, the errors on conserved quantities, such as the energy or the canonical angular momentum when the system is axisymmetric, are bounded for an infinite time (the error on those quantities is second order in ∆t as is the scheme).…”