Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{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-$ \mathsf{E} $ SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and Legendre-Gauss (LG) 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-$ \mathsf{E} $ operators, whose accuracy and maximum timestep restrictions are studied numerically.

翻译：多维对角范数求和-分部（SBP）算子，即具有共置体积点和面点的对角-$ \mathsf{E} $算子，从效率角度出发，对于熵稳定离散化具有吸引力。然而，这类算子的数量有限，且由于缺乏合适的求积规则，现有算子通常在给定多项式阶数下具有相对较高的节点数。我们提出了几种新的三角形和四面体上的对称正权求积规则，适用于构造对角-$ \mathsf{E} $ SBP算子。对于三角形，导出了从一到二十次、面点对应于勒让德-高斯-洛巴托（LGL）和勒让德-高斯（LG）求积规则的求积规则。对于四面体，给出了从一到十次求积规则及其对应的面求积规则。所有求积规则均提供于补充数据仓库中。这些求积规则用于构造新型SBP对角-$ \mathsf{E} $算子，并通过数值手段研究了其精度和最大时间步长限制。