Nonlocal PDF methods for Langevin equations with colored noise

Abstract

Langevin equations describe systems driven by internally generated or externally imposed random excitations. If these excitations correspond to Gaussian white noise, it is relatively straightforward to derive a closed form equation for the joint probability density function (PDF) of state variables. Many natural phenomena present however correlated (colored) excitations. For such problems, a full probabilistic characterization through the resolution of a PDF equation can be obtained through two levels of approximations: first, mixed ensemble moments have to be approximated to lead to a closed system of equations and, second, the resulting nonlocal equations should be at least partially localized to ensure computational efficiency. We propose a new semi-local formulation based on a modified large-eddy diffusivity (LED) approach; the formulation retains most of the accuracy of a fully nonlocal approach while presenting the same order of algorithmic complexity as the standard LED approach. The accuracy of the approach is successfully tested against Monte Carlo simulations.