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

Banff International Research Station for Mathematical Innovation and Discovery

In this talk I will discuss the construction of approximate solutions for the Non-linear Fokker-Planck equation. We utilize the $L^2$-Wasserstein gradient flow structure of this PDEs to perform a semi discretization in time by means of the variational BDF2 method. Our approach can be considered as the natural second order analogue of the Minimizing Movement or JKO scheme. In comparison to our own recent work on constructing solutions to $\lambda$-contractive gradient flows in abstract metric spaces, the technique presented here exploits the differential structure of the underlying $L^2$-Wasserstein space. We directly prove that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation without using the abstract theory of curves of maximal slope. Additionally, we provide strong $L^m$ convergence instead of merely weak convergence in the $L^2$-Wasserstein topology of the time-discrete approximations.

Mathematics; Partial differential equations; Statistical mechanics, structure of matter

video/mp4

Author affiliation: Technical University of Munich

2018-10-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67511

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/