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Generalized Summation-By-Parts Methods: Coordinate Transformations, Quadrature Accuracy, and Functional Superconvergence

We investigate coordinate transformations, quadrature accuracy, and functional superconvergence for diagonal-norm tensor-product generalized summation-by-parts operators. We show that projection operators of degree r ≥ 2p are required to preserve quadrature accuracy, and therefore functional superconvergence, in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summation-byparts operator that is used to approximate the flux terms and (2) the degree of the generalized summation-by-parts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendre-Gauss-Lobatto and Legendre-Gauss element-type operators are considered. When the aforementioned condition (2) is violated for the Legendre-Gauss operators, there is an even-odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the projection operators that correspond to the odd-degree Legendre-Gauss operators.