A finite element/Fourier treatment of the Fokker–Planck equation

A method is proposed for a finite element/Fourier solution of the Fokker–Planck (FP) equation describing Coulomb collisions between particles in a fully ionized, spatially homogeneous plasma. A linearized FP equation is obtained by assuming collisions between test particles and a static background are more important than between the test particles themselves. A full 3D velocity space dependence is maintained using cylindrical coordinates ( v ‖ , v ⊥ , γ ) . When a magnetic field exists, v ‖ is aligned with it and γ corresponds to gyroangle. Distribution functions are approximated by a Fourier representation in the azimuthal angle, γ , and by a 2D finite element representation in the parallel and perpendicular directions. The FP equation can be solved in a fully implicit manner allowing large, stable timesteps and simulations that arrive quickly at equilibrium solutions. The results of several test problems are discussed including a calculation of the resistivity of a Lorentz plasma, the heating and cooling of a test particle distribution, the slowing down of a beam of test particles and the acquisition of a perpendicular flow for a non-flowing Maxwellian test distribution. Robust convergence upon refinement of the finite element/Fourier representation is highlighted.