Why summation by parts is not enough

TL;DR

This paper critically examines the construction of summation-by-parts (SBP) operators, revealing limitations of existing methods and proposing improved design criteria and optimization techniques for more accurate and stable numerical discretizations of PDEs.

Contribution

It identifies key conditions for SBP operators to ensure accuracy and stability, and integrates these into an optimization-based construction framework for arbitrary grids.

Findings

Enhanced accuracy of SBP operators demonstrated through numerical experiments

Identification of conditions for SBP operators to achieve stability and accuracy

Integration of design criteria into an optimization procedure for function space SBP operators

Abstract

We investigate the construction and performance of summation-by-parts (SBP) operators, which offer a powerful framework for the systematic development of structure-preserving numerical discretizations of partial differential equations. Previous approaches for the construction of SBP operators have usually relied on either local methods or sparse differentiation matrices, as commonly used in finite difference schemes. However, these methods often impose implicit requirements that are not part of the formal SBP definition. We demonstrate that adherence to the SBP definition alone does not guarantee the desired accuracy, and we identify conditions for SBP operators to achieve both accuracy and stability. Specifically, we analyze the error minimization for an augmented basis, discuss the role of sparsity, and examine the importance of nullspace consistency in the construction of SBP operators. Furthermore, we show how these design criteria can be integrated into a recently proposed optimization-based construction procedure for function space SBP (FSBP) operators on arbitrary grids. Our findings are supported by numerical experiments that illustrate the improved accuracy for the numerical solution using the proposed SBP operators.