Implicit Schemes for the Fokker-Planck-Landau Equation

We propose time implicit schemes to solve the homogeneous Fokker--Planck--Landau equation in both the isotropic and 3-dimensional geometries. These schemes have properties of conservation and entropy. Moreover, they allow for large time steps, making them faster than the usual explicit schemes. To solve the involved linear systems, we prove that the use of Krylov-like solvers preserves the conservation properties. We show in particular that the conjugate gradient method can be used. Numerical tests are performed for the isotropic case and demonstrate an important gain in terms of CPU time, with the same accuracy as explicit schemes. This work is a first step to the development of fast implicit schemes to solve a class of inhomogeneous kinetic equations.

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