The summation-by-parts (SBP) property provides a rigorous means of proving linear and nonlinear stability. Recently, the SBP property has been extended from tensor-product nodal distributions on multiblock curvilinear meshes to unstructured meshes on arbitrary polytopes. The objective of this paper is to search for efficient cubature rules on quadrilateral elements and perform a comparative analysis of their properties relative to traditional tensor-product operators. To this end, an algorithm for the constrained numerical optimization of multidimensional SBP operators on quadrilateral elements is presented. Using this algorithm, operators are optimized relative to an objective function which accounts for accuracy of the SBP derivative operator. Additionally, properties which affect time stability for explicit time integration methods and conditioning of the node set are calculated and analyzed. Properties necessary to preserve the SBP property are enforced through linear and nonlinear constraints. Numerical experiments are presented comparing tensor-product element-type operators on Legendre-Gauss (LG) and Legendre-Gauss-Lobatto (LGL) nodal distributions and non-tensor-product nodal distributions in order to understand the relative accuracy and computational efficiency of the corresponding methods. It is found that the non-tensor-product nodal distributions are able to achieve cubature rules with lower cubature truncation error compared to tensor-product cubature rules, with fewer nodes. Additionally, the SBP operators constructed on the non-tensor-product cubature nodes are found to have equal or better solution error and efficiency for test cases performed with the linear advection and Euler equations on curvilinear grids.
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