Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs

TL;DR

This paper introduces a new micro-macro stochastic Galerkin method for solving nonlinear Fokker-Planck equations with random inputs, accurately capturing long-term uncertainty effects in large systems.

Contribution

It develops an equilibrium-preserving scheme that improves accuracy over standard stochastic Galerkin methods for uncertain nonlinear Fokker-Planck equations.

Findings

Accurately describes uncertainty-dependent large time behavior.

Outperforms standard methods in numerical tests.

Effective in modeling collective social and biological dynamics.

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 has often to face with physical forces having a significant random component or with particles living in a random environment which 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 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.