Decay of entropy from a conservative spectral method for Fokker-Planck-Landau type equations

Abstract

Expanding upon the conservative spectral method for solving the Landau equation, developed by Zhang and Gamba, a deterministic scheme has been developed for modeling Fokker-Planck-Landau type equations with Maxwell molecules and hard sphere interactions. The original case, corresponding to the classical physical problem of Coulomb interactions, is also included and the stability for all three scenarios investigated. The power of the method is exemplified through simulations demonstrating the decay of relative entropy for both Coulomb interactions and hard potentials. The Coulomb interaction example shows that there is a degenerate spectrum, with the relative entropy decaying at a rate close to the law of two thirds as predicted by Strain and Guo, while the hard potential example exhibits a spectral gap.Expanding upon the conservative spectral method for solving the Landau equation, developed by Zhang and Gamba, a deterministic scheme has been developed for modeling Fokker-Planck-Landau type equations with Maxwell molecules and hard sphere interactions. The original case, corresponding to the classical physical problem of Coulomb interactions, is also included and the stability for all three scenarios investigated. The power of the method is exemplified through simulations demonstrating the decay of relative entropy for both Coulomb interactions and hard potentials. The Coulomb interaction example shows that there is a degenerate spectrum, with the relative entropy decaying at a rate close to the law of two thirds as predicted by Strain and Guo, while the hard potential example exhibits a spectral gap.