Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

In this section, we will discuss two methods that utilize simplices. Our premise is that simplex methods have specific applications where they can be beneficial. While there is ongoing debate regarding the choice of mesh, our focus is on demonstrating the utility of simplex methods.

We observed that higher-order methods applied to quadrilaterals and hexahedra exhibit significantly greater efficiency compared to those on simplices. Our objective is to bridge this efficiency gap. To achieve this, we have sought inspiration from the tensor product structure found in quadrilaterals and hexahedra, aiming to enhance the efficiency of simplex methods by leveraging this structure.

A key aspect of our approach is the Summation by Parts (SBP) property, which is essential for ensuring provable entropy stability. We are committed to developing efficient operators on simplices that incorporate the SBP property. In the following discussion, we will explore the advantages of the SBP property and its implications for our work.