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 [27] 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.
Highlights
This article is concerned with the proof of well-posedness and convergence of a formally higher-order semi-discretization in time, inspired by the Backward Differentiation Formula 2 (BDF2), applied to the non-linear FokkerPlanck equation with no-flux boundary condition:
We proposed in our own recent work [27] a different variational formulation of a semi-discretization in time, i.e., of the Backward Differentiation Formula 2 (BDF2) method
In the proof it is sufficient to have the relaxed averaged weak integral equicontinuity, given in the theorem above
Summary
This article is concerned with the proof of well-posedness and convergence of a formally higher-order semi-discretization in time, inspired by the Backward Differentiation Formula 2 (BDF2), applied to the non-linear FokkerPlanck equation with no-flux boundary condition:. We consider (1) as an evolutionary equation in the space of probability measures P2(Ω) with finite second moment (i.e M2(μ) := Ω x 2 dμ(x) < ∞), where Ω = Rd or Ω ⊂ Rd is a compact domain with Lipschitz-continuous boundary ∂Ω and normal derivative n. The modern approach towards the theoretical analysis of equation (1) is the gradient flow structure in the L2-Wasserstein space (P2(Ω), W2), see [2, 18, 30, 33, 2010 Mathematics Subject Classification. Second order scheme, BDF2, minimizing movements, non-linear diffusion equations. The author would like to thank Daniel Matthes for helpful discussions and remarks
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