Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs

Abstract

Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 527-560, March 2024.
Abstract. Nonlinear Fokker–Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker–Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker–Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.