Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators

Abstract

Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-\( \textsf{E}\) operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However, there is a limited number of such operators, and those currently in existence often have a relatively high node count for a given polynomial order due to a scarcity of suitable quadrature rules. We present several new symmetric positive-weight quadrature rules on triangles and tetrahedra that are suitable for construction of diagonal-\( \textsf{E}\) SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto and Legendre-Gauss quadrature rules are derived. For tetrahedra, quadrature rules of degree one through ten are presented along with the corresponding facet quadrature rules. All of the quadrature rules are provided in a supplementary data repository. The quadrature rules are used to construct novel SBP diagonal-\( \textsf{E}\) operators, whose accuracy and maximum time-step restrictions are studied numerically.