Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

TL;DR

This paper proves that certain finite difference schemes for hyperbolic PDEs with boundary conditions are stable under weak conditions, confirming a conjecture about the sufficiency of the Uniform Kreiss-Lopatinskii Condition.

Contribution

It establishes power boundedness of finite rank perturbations of Toeplitz operators under specific spectral assumptions, advancing understanding of stability in hyperbolic PDE discretizations.

Findings

Finite difference schemes are power bounded under weak spectral conditions.

Supports the conjecture that the Uniform Kreiss-Lopatinskii Condition suffices for stability.

Provides a theoretical foundation for boundary condition stability analysis.

Abstract

In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called Uniform Kreiss-Lopatinskii Condition is sufficient to imply power boundedness.