English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The focus of the present research is on the analysis of local energystability of high-order (including split-form) summation-by-parts methods, withe.g. two-point entropy-conserving fluxes, approximating non-linear conservationlaws. Our main finding is that local energy stability, i.e., the numericalgrowth rate does not exceed the growth rate of the continuous problem, is notguaranteed even when the scheme is non-linearly stable and that this may haveadverse implications for simulation results. We show that entropy-conservingtwo-point fluxes are inherently locally energy unstable, as they can bedissipative or anti-dissipative. Unfortunately, these fluxes are at the core ofmany commonly used high-order entropy-stable extensions, including split-formsummation-by-parts discontinuous Galerkin spectral element methods (or spectralcollocation methods). For the non-linear Burgers equation, we furtherdemonstrate numerically that such schemes cause exponential growth of errorsduring the simulation. Furthermore, we encounter a similar abnormal behaviourfor the compressible Euler equations, for a smooth exact solution of a densitywave. Finally, for the same case, we demonstrate numerically that othercommonly known split-forms, such as the Kennedy and Gruber splitting, are alsolocally energy unstable.

Stability issues of entropy-stable and/or split-form high-order schemes