Approximating Analytic Spectra of Hyperbolic Systems with Summation-by-Parts Finite Difference Operators

TL;DR

This paper investigates how well high-order finite difference methods approximate the spectra of hyperbolic PDEs, highlighting challenges in capturing physical instabilities and proposing best practices for spectral approximation.

Contribution

It demonstrates the effectiveness of summation-by-parts finite difference operators in spectral approximation and discusses the limitations of mesh refinement alone for stability analysis.

Findings

Mesh refinement improves spectral accuracy.

Strict stability is not always achieved.

Eigenvalues may be non-physical despite refinement.

Abstract

In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to accurately discover sources of physical instabilities. By considering the perturbed equations that arise in linearized problems, we study systems in which a lower-order term can act as a source of internal energy within the system. We apply high-order accurate summation-by-parts finite difference operators, with weak enforcement of boundary conditions through the simultaneous-approximation-term technique, which leads to a provably stable numerical discretization with formal order of accuracy given by $p = 2, 3, 4$ and $5$. We derive analytic solutions using Laplace transform methods, which provide important ground truth to ensure numerical convergence at the correct theoretical rate. We derive the analytic spectrum and find that it is better captured with mesh refinement, although dissipative strict stability (where the growth rate of the discrete problem is bounded above by the analytic) is not obtained. We also find that sole reliance on mesh refinement can be a problematic means for determining physical growth rates as some eigenvalues emerge (and persist with mesh refinement) based on spatial order of accuracy but are non-physical. We suggest that numerical methods be used to approximate the spectra when numerical stability is guaranteed and convergence of the numerical spectra is evident with both mesh refinement and increasing order of accuracy.