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


Author(s): Maas, Jan; Matthes, Daniel
Article Title: Long-time behavior of a finite volume discretization for a fourth order diffusion equation
Affiliation IST Austria
Abstract: We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.
Keywords: Wasserstein metric; fourth order equation; gradient flow; structure preserving discretization
Journal Title: Nonlinearity
Volume: 29
Issue 7
ISSN: 1361-6544
Publisher: IOP Publishing Ltd.  
Date Published: 2016-06-10
Start Page: 1992
End Page: 2023
Sponsor: This research was supported by the DFG Collaborative Research Centers TRR 109, ‘ Discretization in Geometry and Dynamics ’ and 1060 ‘ The Mathematics of Emergent Effects ’ .
URL:
DOI: 10.1088/0951-7715/29/7/1992
Open access: yes (repository)