# Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions

## TL;DR

This paper develops structure-preserving stochastic Galerkin methods for Fokker-Planck equations with uncertainties and background interactions, ensuring physical properties and high accuracy in transient and long-term regimes.

## Contribution

It introduces novel stochastic Galerkin schemes that preserve key physical properties for Fokker-Planck equations with background interactions, applicable to collective behavior models.

## Key findings

- Methods preserve nonnegativity, entropy dissipation, and asymptotic behavior.
- Achieve second order accuracy in transient regimes and high order for large times.
- Successfully applied to models with fixed and dynamic background distributions.

## Abstract

This paper is devoted to the construction of structure preserving stochastic Galerkin schemes for Fokker-Planck type equations with uncertainties and interacting with an external distribution, that we refer to as a background distribution. The proposed methods are capable to preserve physical properties in the approximation of statistical moments of the problem like nonnegativity, entropy dissipation and asymptotic behaviour of the expected solution. The introduced methods are second order accurate in the transient regimes and high order for large times. We present applications of the developed schemes to the case of fixed and dynamic background distribution for models of collective behaviour.