2015/09/04 by Horst Osberger, Osberger, Horst, Daniel Matthes +1
Engineering · Physics and Astronomy · #35A15 #35G31 #65M12 #Dynamical Systems (math.DS) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Lattice Boltzmann Simulation Studies #Numerical Analysis (math.NA) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1509.01513
openalex publication_date 2015/09/04 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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 L2-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.