Summation-By-Parts Operators and High-Order Quadrature
For researchers using SBP operators in numerical PDEs, this work clarifies and formalizes the quadrature accuracy of SBP weight matrices, which is important for energy-stable discretizations.
This paper shows that summation-by-parts (SBP) weight matrices are quadrature rules with accuracy matching the difference operator, and this accuracy extends to curvilinear domains when the Jacobian is approximated with the same SBP operator. This ensures accurate approximation of the L2 norm and divergence theorem in SBP discretizations.
Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. 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. 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; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.