Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh

TL;DR

This paper introduces a third-order discontinuous Galerkin method for convection-diffusion equations on unstructured triangular meshes that guarantees the maximum principle without mesh restrictions, maintaining accuracy and stability.

Contribution

The paper presents a novel third-order maximum-principle-satisfying DG method for unstructured meshes, with a new flux calculation ensuring the maximum principle without geometric constraints.

Findings

Method achieves third-order accuracy.

Ensures solutions satisfy maximum principle.

Works on meshes with obtuse triangles.

Abstract

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19, 21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair $(\beta_0,\beta_1)$ in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.