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

Zheng Sun Jose A. Carrillo Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25022

Journal of Computational Physics, 352, 76-104, 2018
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.
discontinuous Galerkin method, positivity-preserving, entropy-entropy dissipation relationship, nonlinear parabolic equation, gradient flow
[ Download ] [ 2018-04-16 10:44:26 uploaded by chiwangshu ] [ 590 downloads ] [ 0 comments ]
@inproceedings{zheng2018a,
  title={A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials},
  author={Zheng Sun, Jose A. Carrillo, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104426147065057},
  booktitle={Journal of Computational Physics},
  volume={352},
  pages={76-104},
  year={2018},
}
Zheng Sun, Jose A. Carrillo, and Chi-Wang Shu. A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. 2018. Vol. 352. In Journal of Computational Physics. pp.76-104. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104426147065057.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved