Alex Bercik

Alex Bercik, David A. Craig Penner, David W. Zingg

The construction of stable, conservative, and accurate volume dissipation is extended to discretizations that possess a generalized summation-by-parts (SBP) property within a tensor-product framework. The dissipation operators can be applied to any finite-difference or spectral-element scheme that uses the SBP framework, including high-order entropy-stable schemes. Additionally, we clarify the incorporation of a variable coefficient within the operator structure and analyze the impact of a boundary correction matrix on operator structure and accuracy. Following the theoretical development and construction of novel dissipation operators, we relate the presented volume dissipation to the use of upwind SBP operators. When applied to spectral-element methods, the presented approach yields unique dissipation operators that can also be derived through alternative approaches involving orthogonal polynomials. Numerical examples featuring the linear convection, Burgers, and Euler equations verify the properties of the constructed dissipation operators and assess their performance compared to existing upwind SBP schemes, including linear stability behaviour. When applied to entropy-stable schemes, the presented approach results in accurate and robust methods that can solve a broader range of problems where comparable existing methods fail.

Alex Bercik, David W. Zingg

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.