Author(s):

Maas, Jan; Matthes, Daniel

Article Title: 
Longtime behavior of a finite volume discretization for a fourth order diffusion equation

Affiliation 
IST Austria 
Abstract: 
We consider a nonstandard finitevolume discretization of a strongly nonlinear fourth order diffusion equation on the ddimensional 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 FokkerPlanck 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 discretetocontinuous limit. Using the dissipation, we derive estimates on the longtime 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:

13616544

Publisher:

IOP Publishing Ltd.

Date Published:

20160610

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: 
10.1088/09517715/29/7/1992

Open access: 
yes (repository) 