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

2015/01/20 by Horst Osberger, Osberger, Horst · 1 citation
Mathematics · Engineering · #Geometric Analysis and Curvature Flows #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1501.04800

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 L²-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.

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