In this work we explore the fidelity of numerical approximations to continuous spectra of hyperbolic partial differential equation systems. We are particularly interested in the ability of discrete methods to accurately discover sources of physical instabilities. By focusing on the perturbed equations that arise in linearized problems, we apply high-order accurate summation-by-parts finite difference operators, with weak enforcement of boundary conditions through the simulataneous-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 for ensuring numerical convergence at the correct theoretical rate. We find that the continuous spectrum is better captured with mesh refinement, although dissipative strict stability (where the growth rate of the discrete problem is bounded above by the continuous) is not obtained. However, 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 discrete spectra when numerical stability is guaranteed and convergence of the discrete spectra is evident with both mesh refinement and increasing order of accuracy.

翻译：本文探讨了双曲偏微分方程系统连续谱数值逼近的保真度。我们特别关注离散方法准确发现物理不稳定性来源的能力。通过聚焦于线性化问题中出现的扰动方程，我们应用高阶精度的求和分部有限差分算子，并借助同步逼近项技术弱实施边界条件，从而构建出可证明稳定的数值离散格式，其形式精度阶数为 $p = 2, 3, 4$ 和 $5$。我们采用拉普拉斯变换方法推导解析解，这为确保数值以正确理论速率收敛提供了重要的基准真值。研究发现：虽然无法获得耗散严格稳定性（即离散问题的增长率始终受连续问题增长率上界约束），但通过网格细化能更好地捕捉连续谱。然而，我们也发现单纯依赖网格细化来确定物理增长率可能存在缺陷，因为某些特征值会基于空间精度阶数出现（并在网格细化过程中持续存在）但实际并无物理意义。我们建议：当数值稳定性得到保证，且离散谱能随网格细化和精度阶数提升呈现明显收敛趋势时，可采用数值方法逼近离散谱。