Summation-By-Parts Operators and High-Order Quadrature

TL;DR

This paper explores the relationship between summation-by-parts (SBP) operators and high-order quadrature rules, demonstrating how SBP weight matrices relate to trapezoid rules and extend to curvilinear domains, impacting energy-stable PDE discretizations.

Contribution

It establishes a connection between SBP weight matrices and trapezoid rules with end corrections, showing their accuracy and extension to curvilinear domains, enhancing SBP-based discretizations.

Findings

SBP weight matrices are related to trapezoid rules with end corrections.

SBP quadrature accuracy extends to curvilinear domains with Jacobian approximation.

Discrete SBP norm accurately approximates the L2 norm.

Abstract

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.