An Adjoint-based Numerical Method for a class of nonlinear Fokker-Planck Equations

TL;DR

This paper presents an adjoint-based numerical method for nonlinear Fokker-Planck equations, leveraging their relationship with Hamilton-Jacobi equations to ensure properties like positivity and mass preservation.

Contribution

The authors develop a novel numerical scheme for Fokker-Planck equations by utilizing their adjoint relationship with Hamilton-Jacobi equations, ensuring desirable numerical properties.

Findings

The method preserves positivity and mass in simulations.

Applications include mean-field games and crowd motion models.

Numerical results demonstrate the scheme's effectiveness.

Abstract

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.