On an eigenvalue property of Summation-By-Parts operators

TL;DR

This paper investigates the eigenvalue property of Summation-By-Parts (SBP) operators, showing it doesn't hold universally, but can be fixed with small perturbations, and that pseudospectral methods always satisfy it.

Contribution

The paper proves three key results about the eigenvalue property of SBP operators, clarifying its limitations and how to ensure it holds.

Findings

Not all nullspace consistent SBP operators have the eigenvalue property.

Adding a small perturbation can restore the eigenvalue property without losing accuracy.

All pseudospectral methods satisfy the eigenvalue property.

Abstract

Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property.