A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker–Planck operator that can be inverted by the conjugate-gradient method. We compare the effects of different penalization operators, and conclude that the Fokker–Planck (FP) operator is a good choice. Such schemes overcome the stiffness of the collision operator in the fluid regime, and can capture the fluid dynamic limit without numerically resolving the small Knudsen number. Numerical experiments demonstrate that the schemes possess the AP property for general initial data, with numerical accuracy uniformly in the Knudsen number.