Summation-by-parts operators and high-order quadrature

Authors:
J. E. Hicken;D. W. Zingg
Affiliations:
-;-
Venue:
Journal of Computational and Applied Mathematics
Year:
2013

Citing 9
Cited 6

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Dual consistency and functional accuracy: a finite-difference perspective

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Duality based boundary conditions and dual consistent finite difference discretizations of the Navier-Stokes and Euler equations

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Abstract

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the L^2 norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem.