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

The Multidimensional SBP operator is designed to approximate derivatives with a specified level of accuracy. It must be decomposable with the SPD matrix, and it must satisfy the condition Q + Q transpose equals E, where E approximates the surface integral.

In the context of multidimensional operators on simplices, we categorize them into three classes. The first class, SBP-Ω, features distinct volume quadrature nodes represented by circles and facet quadrature nodes represented by orange squares. In the second class, SBP-Γ, some volume quadrature nodes are located on the facets, but they do not necessarily coincide with the facet quadrature nodes. The third class, SBP-E, has volume quadrature nodes on the facets that coincide with the facet quadrature nodes, leading to a diagonal matrix that can enhance computational efficiency.

While the previous discussion focused on linear convection and its linear stability, our interest now shifts to nonlinear stability. The slide presents the equation for entropy, expressed as an equality for smooth solutions. This auxiliary equation must be satisfied in addition to the conservation equations, which facilitates various proofs.