On the Practical Impact of Local Linear Instabilities in Entropy-Stable Schemes

Local linear instability refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous linearized problem. In the context of nonlinear entropy-stable discretizations, we argue that local linear instabilities should be interpreted as a source of numerical error whose practical impact is often negligible compared with other discretization errors. For split-form discretizations of the variable-coefficient linear advection equation, such as those resulting from linearizations of entropy-stable discretizations of the Burgers equation, perturbations can indeed exhibit unphysical modal growth. However, we demonstrate that this growth satisfies physically interpretable bounds and is typically small. Furthermore, through modified-equation analysis and numerical experiments, we show that the growth is dominated by highly oscillatory and boundary-localized unphysical modes, and can therefore be readily controlled by small amounts of numerical dissipation. More generally, this modal perturbation growth does not extend directly to nonlinear two-point-flux discretizations of the type used in entropy-stable discretizations of the Euler equations. Floquet analysis demonstrates that unstable spectra of frozen-baseflow Jacobians need not lead to unstable perturbation growth. Using the geometric flux for the variable-coefficient linear advection equation, we derive a sharp perturbation growth bound predicting negligible growth, then show analogous behaviour for the logarithmic flux numerically. Finally, we argue that robustness issues observed for entropy-stable schemes in density-wave problems are better attributed to poor near-vacuum behaviour of the logarithmic mean than to local linear instabilities. Overall, our results suggest that local linear instabilities do not pose a practical obstacle to the use of high-order entropy-stable schemes.