Subcycling of particle orbits in variational, geometric electromagnetic particle-in-cell methods
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This paper introduces two subcycling algorithms for particle orbits in variational, geometric particle-in-cell methods that address the Vlasov-Maxwell system in magnetized plasmas. The purpose of subcycling is to allow for time steps in the global field solves at or longer than the gyroperiod time scale while sampling the local particle cyclotron orbits accurately. Both algorithms retain the electromagnetic gauge invariance of the discrete action, guaranteeing a local charge conservation law, and the variational approach provides a bounded long-time energy behaviour. In the first algorithm, the global field solves are explicit and the local particle push implicit for each particle individually. The requirement for gauge invariance leads to a peculiar subcycling scheme where the magnetic field is orbit-averaged and the effect of the electric field on the particle orbits is evaluated once during the sybcycling period. Numerical tests with this algorithm indicate that artificial oscillations may occur if the electric field impulse on the particles grows large. The oscillations are observed to vanish if orbit-averaging is enforced also for the electric field but then the variational particle push is lost. The second algorithm, a fully implicit one, is proposed to remedy the possible issues of the semi-explicit algorithm. It is observed that both magnetic and electric field can be orbit-averaged, the gauge invariance and consequently the charge conservation retained, while the algorithm remains variational. These requirements, however, appear to require a fully implicit approach. Numerical experiments with and adaptive time-step control for the implicit scheme are left for a future study.
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