Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh
For computational scientists solving convection-diffusion equations, this work provides a high-order numerical scheme that guarantees physical bounds without mesh quality constraints.
This paper develops third-order maximum-principle-satisfying direct discontinuous Galerkin methods for convection-diffusion equations on unstructured triangular meshes. The method ensures the quadratic polynomial solution satisfies the strict maximum principle while maintaining third-order accuracy, with no geometric restrictions on the mesh.
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 $(β_0,β_1)$ 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.