On the Practical Impact of Local Linear Instabilities in Entropy-Stable Schemes
For researchers using high-order entropy-stable schemes, this paper clarifies that local linear instabilities are not a practical obstacle, addressing a concern about their reliability.
The paper argues that local linear instabilities in entropy-stable schemes are a source of numerical error with negligible practical impact, as they are bounded, dominated by oscillatory modes, and controllable via dissipation. It shows that such instabilities do not extend to nonlinear discretizations and are not the cause of robustness issues in density-wave problems.
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.