A BDF2-Approach for the Non-linear Fokker-Planck Equation

TL;DR

This paper proves convergence of a BDF2 variational scheme for the non-linear Fokker-Planck equation, extending previous results by removing convexity assumptions and establishing strong convergence to weak solutions.

Contribution

It introduces a novel BDF2-based variational approach that does not require uniform semi-convexity and directly proves strong convergence to weak solutions.

Findings

Proves convergence of the BDF2 scheme for the non-linear Fokker-Planck equation.

Establishes strong convergence of time-discrete approximations.

Shows the limit curve is a weak solution without relying on abstract slope theory.

Abstract

We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying $L^2$-Wasserstein space. The technique presented here extends and strengthens the results of our own recent work on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the time-discrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.