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Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker–Planck equations
Numerische Mathematik ( IF 2.1 ) Pub Date : 2019-12-18 , DOI: 10.1007/s00211-019-01094-y
Guillaume Dujardin , Frédéric Hérau , Pauline Lafitte

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker–Planck equations. In particular, for these discretizations 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.

中文翻译:

离散 Fokker-Planck 方程的矫顽力、低矫顽力、指数时间衰减和模拟

在本文中,我们提出并研究了齐次和非齐次一维 Fokker-Planck 方程的几个离散版本。特别是,对于速度和空间的这些离散化,我们证明了对时间连续方程和时间离散方程解的平衡的指数收敛。我们的方法使用新型离散庞加莱不等式对速度导数进行“双向”离散化。对于非同质问题,我们将虚伪方法应用于离散情况。
更新日期:2019-12-18
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