Summation-by-parts operators for correction procedure via reconstruction

TL;DR

This paper reformulates the correction procedure via reconstruction (CPR) methods using summation-by-parts (SBP) operators, providing proofs of conservation and stability, including entropy stability for Burgers' equation, thus advancing the theoretical foundation of high-order numerical schemes for conservation laws.

Contribution

It introduces an SBP-based reformulation of CPR methods, enabling rigorous proofs of conservation, linear stability, and entropy stability for a broad class of high-order schemes.

Findings

Reformulation of CPR methods using SBP operators.

Proofs of conservation and stability in discrete norms.

Entropy stability established for Burgers' equation.

Abstract

The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.