Improved Particle Time Integration for Solving the Vlasov-Fokker-Planck Equation

In this study, we extend the previous accurate particle time integration scheme of the Vlasov-Fokker-Planck equation to accommodate spatially inhomogeneous electromagnetic fields. The new approach stems from an improvement of time-symmetry by introducing an iterative mid-step correction. Further, we adopted an additive correction to the electric field to assure consistency with the negative gradient of the electric potentials in the direction of the particle displacement. Therefore, the proposed approach ensures strict conservation of the Hamiltonian structure and achieves accurate time integration of the particle position and velocity. In addition, a linear drift and a constant diffusion coefficient are considered to deal with Coulomb collisions. The validation of the scheme is performed first for the 2D dynamics of a charged particle. Our method shows that an increase of mid-step evaluations improves time symmetry. Moreover, additive correction guarantees conservation of invariants, as total energy and angular momentum. Therefore, time step sizes beyond the gyro-frequency limits are realizable. In addition, collisional test cases are considered and the results are compared to those obtained with Direct Simulation Monte Carlo (DSMC). It can be observed that our method still is accurate for large time steps beyond the gyro- and collision-frequency limits.