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

## TL;DR

This paper introduces a high-order discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems, ensuring positivity and entropy decay, with proven accuracy and applicability to multi-dimensional cases.

## Contribution

The paper develops a novel high-order discontinuous Galerkin scheme that preserves positivity and entropy decay for a broad class of nonlinear parabolic equations with interaction potentials.

## Key findings

- The scheme achieves high-order accuracy on smooth problems.
- It preserves non-negativity of solutions with a positivity limiter.
- Numerical tests confirm effectiveness for long-time behavior.

## 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.