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

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.

Authors:
; ;
Publication Date:
Research Org.:
Brown Univ., Providence, RI (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1459035
Alternate Identifier(s):
OSTI ID: 1511584
Grant/Contract Number:  
FG02-08ER25863
Resource Type:
Published Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Name: Journal of Computational Physics Journal Volume: 352 Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Discontinuous Galerkin method; Positivity-preserving; Entropy–entropy dissipation relationship; Nonlinear parabolic equation; Gradient flow

Citation Formats

Sun, Zheng, Carrillo, José A., and Shu, Chi-Wang. A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. United States: N. p., 2018. Web. doi:10.1016/j.jcp.2017.09.050.
Sun, Zheng, Carrillo, José A., & Shu, Chi-Wang. A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. United States. doi:10.1016/j.jcp.2017.09.050.
Sun, Zheng, Carrillo, José A., and Shu, Chi-Wang. Mon . "A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials". United States. doi:10.1016/j.jcp.2017.09.050.
@article{osti_1459035,
title = {A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials},
author = {Sun, Zheng and Carrillo, José A. and Shu, Chi-Wang},
abstractNote = {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.},
doi = {10.1016/j.jcp.2017.09.050},
journal = {Journal of Computational Physics},
number = C,
volume = 352,
place = {United States},
year = {2018},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1016/j.jcp.2017.09.050

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