---
_id: '1261'
abstract:
- lang: eng
  text: '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.'
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 ’ .
article_processing_charge: No
arxiv: 1
author:
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Daniel
  full_name: Matthes, Daniel
  last_name: Matthes
citation:
  ama: Maas J, Matthes D. Long-time behavior of a finite volume discretization for
    a fourth order diffusion equation. <i>Nonlinearity</i>. 2016;29(7):1992-2023.
    doi:<a href="https://doi.org/10.1088/0951-7715/29/7/1992">10.1088/0951-7715/29/7/1992</a>
  apa: Maas, J., &#38; Matthes, D. (2016). Long-time behavior of a finite volume discretization
    for a fourth order diffusion equation. <i>Nonlinearity</i>. IOP Publishing. <a
    href="https://doi.org/10.1088/0951-7715/29/7/1992">https://doi.org/10.1088/0951-7715/29/7/1992</a>
  chicago: Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization
    for a Fourth Order Diffusion Equation.” <i>Nonlinearity</i>. IOP Publishing, 2016.
    <a href="https://doi.org/10.1088/0951-7715/29/7/1992">https://doi.org/10.1088/0951-7715/29/7/1992</a>.
  ieee: J. Maas and D. Matthes, “Long-time behavior of a finite volume discretization
    for a fourth order diffusion equation,” <i>Nonlinearity</i>, vol. 29, no. 7. IOP
    Publishing, pp. 1992–2023, 2016.
  ista: Maas J, Matthes D. 2016. Long-time behavior of a finite volume discretization
    for a fourth order diffusion equation. Nonlinearity. 29(7), 1992–2023.
  mla: Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization
    for a Fourth Order Diffusion Equation.” <i>Nonlinearity</i>, vol. 29, no. 7, IOP
    Publishing, 2016, pp. 1992–2023, doi:<a href="https://doi.org/10.1088/0951-7715/29/7/1992">10.1088/0951-7715/29/7/1992</a>.
  short: J. Maas, D. Matthes, Nonlinearity 29 (2016) 1992–2023.
corr_author: '1'
date_created: 2018-12-11T11:51:00Z
date_published: 2016-06-10T00:00:00Z
date_updated: 2025-09-22T09:00:50Z
day: '10'
department:
- _id: JaMa
doi: 10.1088/0951-7715/29/7/1992
external_id:
  arxiv:
  - '1505.03178'
  isi:
  - '000378862800006'
intvolume: '        29'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1505.03178
month: '06'
oa: 1
oa_version: Preprint
page: 1992 - 2023
publication: Nonlinearity
publication_status: published
publisher: IOP Publishing
publist_id: '6062'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Long-time behavior of a finite volume discretization for a fourth order diffusion
  equation
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 29
year: '2016'
...