An Implicit Energy-Conservative 2D Fokker–Planck Algorithm: I. Difference Scheme

Numerical energy conservation in Fokker–Planck problems requires the energy moment of the Fokker–Planck equation to cancel exactly. However, standard discretization techniques not only do not observe this requirement (thus precluding exact energy conservation), but they also demand very refined meshes to keep the energy error under control. In this paper, a new difference scheme for multidimensional Fokker–Planck problems that improves the numerical cancellation of the energy moment is proposed. Crucial to this new development is the reformulation of the friction term in the Fokker–Planck collision operator using Maxwell stress tensor formalism. As a result, the Fokker–Planck collision operator takes the form of a double divergence operating on a tensor, which is suitable for particle and energy conservative differencing. Numerical results show that the new discretization scheme improves the cancellation of the energy moment integral over standard approaches by at least an order of magnitude.