Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker- Planck equations

Guillaume Dujardin (1) , Frédéric Hérau (2) , Pauline Lafitte-Godillon (3, 4)

1 Paradyse - Systèmes de particules et systèmes dynamiques
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
4 Fédération de Mathématiques de CentraleSupélec

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these dis-cretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a " two-direction " discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

Domaines: Equations aux dérivées partielles [math.AP]

Guillaume Dujardin, Frédéric Hérau, Pauline Lafitte-Godillon. Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker- Planck equations. Numerische Mathematik, 2020, 144, ⟨10.1007/s00211-019-01094-y⟩. ⟨hal-01702545⟩