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

One disadvantage of entropy stable schemes is the high computational cost associated with two-point flux functions. It may seem counterintuitive, but it is necessary to compute the two-point flux between every pair of points within an element. This is where the tensor product structure proves advantageous, as it reduces the penalty associated with multidimensional operators.

For a polynomial degree ( P = 2 ), consider a single node; it requires a two-point flux calculation with every other node in the element. As the polynomial degree increases to ( P = 4 ) or ( P = 6 ), the number of two-point flux functions grows as ( P^{2D} ). In contrast, for a tensor product operator, such as those used on quadrilaterals or hexahedra, the number of functions scales as ( D + 1 ). This more favorable scaling is one reason why simplex schemes are less efficient.

To address these inefficiencies, my students proposed two strategies that leverage the tensor product structure. The first idea involves using collapse coordinates, while the second suggests converting triangles into quadrilaterals or tetrahedra into hexahedra. I will elaborate on each of these approaches.