Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations

Abstract

A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equation's underlying gradient flow structure w.r.t. the L 2 -Wasserstein distance, and adapts numerous of its most important structural properties by construction, as conservation of mass and entropy-dissipation.In this paper, the long-time behaviour of our discretization is analyzed: We show that discrete solutions decay exponentially to equilibrium at the same rate as smooth solutions of the origin problem. Moreover, we give a proof of convergence of discrete entropy minimizers towards Barenblatt-profiles or Gaussians, respectively, using Γ-convergence. arXiv:1501.04800v2 [math.NA]

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

“…Here we continue in the spirit of [34], where a discretization was performed on grounds of (1)'s gradient flow structure with respect to the L 2 -Wasserstein metric, which leads to a scheme that simultaneously preserves two essential Lyapunov functionals. From these Lyapunov functionals, estimates on the fully discrete solutions were derived and have been used to analyze their long-time asymptotics [39] and the discrete-to-continuous limit [34].…”

“…However, here we do not use the Lagrangian structure behind (1) -which was essential in [34,39] -but define a scheme on grounds of a finite-volume discretization. Our ansatz is motivated by a particular structure-preserving discretization of linear Fokker-Planck equations, which has been introduced simultaneously in [8,31,37].…”

“…Our ansatz is motivated by a particular structure-preserving discretization of linear Fokker-Planck equations, which has been introduced simultaneously in [8,31,37]. Using this "Eulerian approach", we overcome the limitation of [34,39] to d = 1 space dimension. The similarities with [34,39] are that we rely on the gradient flow formulation of (1), and that we design the discretization in such a way that enforces monotonicity of two Lyapunov functionals.…”

14. Convergence of a fully discrete variational scheme for a thin-film equation

“…A comment is in place on related Lagrangian schemes in higher spatial dimensions. Here, and also in our related works [31,28,29], the most significant benefit from working on a onedimensional interval, is that the space of densities is flat with respect to the L 2 -Wasserstein metric; it is of non-positive curvature in higher dimensions, which makes the numerical approximation of the Wasserstein distance significantly more difficult. Just recently, a very promising approach for a truely structure-preserving discretization in higher space dimensions has been made [3].…”

“…central finite-difference discretizations. Another one, having different properties, is studied in [31]. Our convergence result only applies to the particular form (12), since only for that one, we obtain "the right" Lyapunov functionals that provide the a priori estimates for the discrete-to-continuous limit.…”