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

This presentation reviews two efficient new nonlinearly-stable high-order methods applicable to simplices, both of which have the summation-by-parts property where integration-by-parts is mimicked discretely, which enables proofs of entropy stability. Both methods exploit a tensor-product structure on simplices to achieve efficiencies on simplices comparable to methods with a tensor-product structure developed for quadrilaterals and hexahedra. In the first approach presented, the tensor-product form is achieved through a collapsed coordinate transformation, an idea that has been used in the past. Through the development of schemes exploiting this strategy while having the summation-by parts property, entropy stability can be proven, thus producing methods with the same accuracy as existing entropy-stable multidimensional summation-by-parts operators with substantially reduced computational cost. The second new approach, tensor-product split-simplex operators, splits the simplices into quadrilateral or hexahedral subdomains, maps tensor-product summation-by-parts operators onto these subdomains, and reassembles into multidimensional operators using a continuous interface formulation. This approach is readily used in an entropy-stable framework as a result of the summation-by-parts property and also leads to substantial efficiency improvements relative to existing multidimensional summation-by-parts operators.