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

The Chandra Gauss nodal distribution is utilized on the reference element for ( p = 4 ). The two-point flux evaluation scale is ( p^{p+1} ). When employing a co-located formulation, we encounter significant time step restrictions due to node spacing. To address this issue, we adopted a modal formulation, which alleviates these constraints.

A key aspect of enhancing efficiency is the weight-adjusted approximation developed by Bessie Chan, which will be discussed by the next speaker.

In the accompanying slide, dashed lines represent entropy conservative fluxes, while solid lines indicate entropy stable fluxes. Although the entropy stable method introduces a slight error, all results are symmetric. The upper section of the slide presents 2D results, while the lower section shows 3D results based on the Euler equations.

The comparison focuses on the collapse coordinate terms of product implementation versus a truly multidimensional approach. The primary takeaway is that both methods yield identical accuracy, as evidenced by the overlapping curves. Thus, we have not sacrificed accuracy by utilizing collapsed coordinates. The next consideration is the associated cost.