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In order to produce the desired global order of accuracy, high-order finite-dierence methods require suitably accurate stable boundary schemes. Various tools can be helpful in providing insight into the stability of numerical boundary schemes, including spectra, pseudospectra, and the singular value decomposition. In this paper, these tools are applied to several dierent discretizations of the linear convection equation which display various types of instability. Pseudospectra are seen to be a convenient and eective means of detecting instabilities. The same instabilities can also be revealed by either the spectrum or the spectrum of the associated circulant matrix resulting from the assumption of periodic boundary conditions. Finally, the singular value decomposition is shown to be a good indicator of the cause of the instability.

On Linear Stability Analysis of High-Order Finite-Difference Methods