An implicit Vlasov–Fokker–Planck code to model non-local electron transport in 2-D with magnetic fields

We describe an implicit finite-difference scheme for solving the Vlasov–Fokker–Planck equation and Maxwell’s equations in 2 spatial dimensions including, for the first time, self-consistent magnetic fields. These equations model the coupled phenomena of magnetic field generation and magnetised electron transport in collisional plasmas, such as laser-produced plasmas, in the non-relativistic limit. The kinetic description of the plasma enables the scheme to properly describe these phenomena in the regime where the temperature and density scale lengths become comparable to the transport mean-free-path. In addition to including the self-consistent magnetic field, other improvements over previous Fokker–Planck codes have been made which result in a robust scheme that can work with a large time step. The scheme employs Cartesian geometry and solves for the electromagnetic field components Ex, Ey and Bz. Extension to all field components and cylindrical geometry is possible.