We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov– Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by t...

We study the existence and the uniqueness of a solution φ to the linear Fokker-Planck equation −∆φ+ div(φF) = f in a bounded domain of R when F is a “confinement” vector field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows. Equation de Fokker-Planck dans un domaine b...

Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on a global averages over stochastic flow. We show how to compute the...

In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21, 23]. We show that the kernel modes that define the spectral method have the correct grazing collis...

The connection between stochastic differential equations and associated Fokker-Planck equations is elucidated by the full functional calculus. One-variable equations with either additive or multiplicative noise are considered. The central focus is on approximate Fokker-Planck equations which describe the consequences of using "colored" noise, which has an exponential correlation function and a ...

We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker– Planck equation featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30: 621–651, 2009) in...

This paper presents an application of multi-scale finite element methods to the solution of the multi-dimensional Fokker–Planck equation. The Fokker–Planck, or forward Kolmogorov, equation is a degenerate convective–diffusion equation arising in Markov-Process theory. It governs the evolution of the transition probability density function of the response of a broad class of dynamical systems dr...

The present Note establishes the self-averaging, radiative transfer limit for the two-frequency Wigner distribution for classical waves in random media. Depending on the ratio of the wavelength to the correlation length the limiting equation is either a Boltzmann-like integral equation or a Fokker–Planck-like differential equation in the phase space. The limiting equation is used to estimate th...

The convergence properties of the stationary Fokker-Planck algorithm for the estimation of the asymptotic density of stochastic search processes is studied. Theoretical and empirical arguments for the characterization of convergence of the estimation in the case of separable and nonseparable nonlinear optimization problems are given. Some implications of the convergence of stationary Fokker-Pla...

We study nonlinear stochastic systems with time-delayed feedback using the concept of delay Fokker-Planck equations introduced by Guillouzic, L'Heureux, and Longtin. We derive an analytical expression for stationary distributions using first-order perturbation theory. We demonstrate how to determine drift functions and noise amplitudes of this kind of systems from experimental data. In addition...