# 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}

}

DOI: 10.1016/j.jcp.2017.09.050

*Citation information provided by*

Web of Science

Web of Science