# THIRD ORDER MAXIMUM-PRINCIPLE-SATISFYING DG SCHEMES Third Order Maximum-Principle-Satisfying DG schemes for Convection-Diffusion problems with Anisotropic Diffusivity DIFFUSIVITY

## TL;DR

This paper develops third order accurate discontinuous Galerkin schemes for convection-diffusion equations with variable diffusivity, ensuring maximum principle preservation and high accuracy on rectangular meshes in 1D, 2D, and 3D.

## Contribution

It introduces a novel DG scheme with a scaling limiter that preserves maximum principles while maintaining third order accuracy for nonlinear convection-diffusion problems.

## Key findings

- The schemes achieve third order accuracy in multiple dimensions.
- Numerical results confirm the maximum principle preservation.
- The methods are effective for nonlinear convection-diffusion equations.

## Abstract

For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection-diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in [Liu and Yu, SIAM J. Sci. Comput. 36(5): A2296{A2325, 2014] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.