A BDF2-Approach for the Non-linear Fokker-Planck Equation
This work provides a theoretical convergence guarantee for a higher-order time discretization of a class of non-linear PDEs, improving upon previous results that required stronger assumptions or yielded weaker convergence.
The authors prove convergence of a BDF2 method for the non-linear Fokker-Planck equation, achieving strong convergence of time-discrete approximations to a weak solution without requiring uniform semi-convexity of the energy functional.
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.