A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials

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Title: A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
Author(s): Sun, Z
Carrillo de la Plata, J
Shu, CW
Item Type: Journal Article
Abstract: We consider a class of time-dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker–Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics.
Publication Date: 28-Sep-2017
Date of Acceptance: 22-Sep-2017
URI: http://hdl.handle.net/10044/1/51233
DOI: https://dx.doi.org/10.1016/j.jcp.2017.09.050
ISSN: 0021-9991
Publisher: Elsevier
Start Page: 76
End Page: 104
Journal / Book Title: Journal of Computational Physics
Volume: 352
Sponsor/Funder: The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: WM120001
Copyright Statement: © 2017 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Keywords: 01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Applied Mathematics
Publication Status: Published
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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