Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians

TL;DR

This paper introduces a generalized framework for summation-by-parts (SBP) operators within high-order conservation law methods, enabling entropy stability on complex grids with dense norms and varying Jacobians.

Contribution

It extends SBP operators to include dense norms and boundary-free points, achieving entropy stability for Burgers' equation and stability on curvilinear grids.

Findings

SBP operators with dense norms enable entropy stable formulations.

Stability achieved on curvilinear grids with varying Jacobians.

Extension of SBP-CPR framework to broader classes of problems.

Abstract

A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable split-form of Burgers' equation. Moreover, overcoming limitations of the finite difference framework, stability for curvilinear grids and dense norms is obtained for SBP CPR methods by using a suitable way to compute the Jacobian.