Diffusive Limit of a Two-Dimensional Well-Balanced Scheme for the Free Klein-Kramers Equation

The Fokker--Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive limit. In order to set up a well-balanced discretization relying on an $S$-matrix, exact steady states of the continuous equation are derived. The ability of the scheme to keep these stationary solutions invariant produces the discretization of the local differential operator which mimics the collision process. The aforementioned scheme can be reformulated as an implicit-explicit one, which is proved to be both well-balanced and asymptotic-preserving in the diffusion limit. Several numerical benchmarks, computed on coarse grids, are displayed so as to illustrate the results.