We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We also present results from numerical experiments.

We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient flow of an entropy functional in the L 2 -Wasserstein metric, the second is the Lagrangian nature, meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps. The resulting numerical scheme is entropy diminishing and mass conserving. Further, the scheme is weakly stable, which allows us to prove convergence under certain regularity assumptions. Finally, we present results from numerical experiments in space dimension d = 2.arXiv:1509.07721v3 [math.NA]

Abstract. A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme.Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate.

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]

Abstract. This paper is concerned with a rigorous convergence analysis of a fully discrete Lagrangian scheme for the Hele-Shaw flow, which is the fourth order thin-film equation with linear mobility in one space dimension. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. Apart from its Lagrangian character -which guarantees positivity and mass conservation -the main feature of our discretization is that it dissipates both the Dirichlet energy and the logarithmic entropy. The interplay between these two dissipations paves the way to proving convergence of the discrete approximations to a weak solution in the discrete-to-continuous limit. Thanks to the time-implicit character of the scheme, no CFL-type condition is needed. Numerical experiments illustrate the practicability of the scheme.