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

J. Maas, D. Matthes, Nonlinearity 29 (2016) 1992–2023.


Journal Article | Published | English
Author
;
Department
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.
Publishing Year
Date Published
2016-06-10
Journal Title
Nonlinearity
Acknowledgement
This research was supported by the DFG Collaborative Research Centers TRR 109, ‘ Discretization in Geometry and Dynamics ’ and 1060 ‘ The Mathematics of Emergent Effects ’ .
Volume
29
Issue
7
Page
1992 - 2023
IST-REx-ID

Cite this

Maas J, Matthes D. Long-time behavior of a finite volume discretization for a fourth order diffusion equation. Nonlinearity. 2016;29(7):1992-2023. doi:10.1088/0951-7715/29/7/1992
Maas, J., & Matthes, D. (2016). Long-time behavior of a finite volume discretization for a fourth order diffusion equation. Nonlinearity, 29(7), 1992–2023. https://doi.org/10.1088/0951-7715/29/7/1992
Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization for a Fourth Order Diffusion Equation.” Nonlinearity 29, no. 7 (2016): 1992–2023. https://doi.org/10.1088/0951-7715/29/7/1992.
J. Maas and D. Matthes, “Long-time behavior of a finite volume discretization for a fourth order diffusion equation,” Nonlinearity, vol. 29, no. 7, pp. 1992–2023, 2016.
Maas J, Matthes D. 2016. Long-time behavior of a finite volume discretization for a fourth order diffusion equation. Nonlinearity. 29(7), 1992–2023.
Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization for a Fourth Order Diffusion Equation.” Nonlinearity, vol. 29, no. 7, IOP Publishing Ltd., 2016, pp. 1992–2023, doi:10.1088/0951-7715/29/7/1992.

Link(s) to Main File(s)
Access Level
OA Open Access

Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar