Abstract
The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.
Highlights
For consistent discretisations of well-posed linear partial differential equations (PDEs), it is well-known that stability is necessary and sufficient for a numerical solution to converge to the analytical solution
This growth may push the solution into a regime where the non-linear terms are no longer negligible. This is the mechanism that produces turbulence or shocks even when energy stable, such as central difference, schemes are used. (In the same way, artificial growth that exceeds the growth of the continuous problem caused by an energy unstable operator may erroneously push the solution into the non-linear regime.) the spectrum of an energy stable difference operator approximating a variable coefficient problem may be positive
The results of the longterm simulations of the entropy-conserving or entropy-dissipative discretisations shown in Figs. 11, 12, 13 and 14 are highly worrisome. They demonstrate that the discretisations might generate results that appear stable, but the simulations are nonphysical as they allow for an exponential growth of small scale fluctuations until the non-linear stability estimate kicks in and provides a global bound
Summary
For consistent discretisations of well-posed linear partial differential equations (PDEs), it is well-known that stability (typically in L2) is necessary and sufficient for a numerical solution to converge to the analytical solution. This was proven by Lax and Richtmyer in [24]. The total specific entropy should only increase in a closed system This translates into the solution satisfying an extra partial differential inequality.
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