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.

翻译：目前研究的重点是分析高顺序(包括分形成型)加在一起法的当地能源稳定性,例如,两点对流调节通量,近似非线性节能法。我们的主要发现是,即使这个办法不线性稳定,而且可能对模拟结果产生不良影响,当地能源稳定性也得不到保障。我们表明,对双点保值通量的两点通量本身就具有本地能源不稳定性,因为它们可能是分解或反分解性的。不幸的是,这些通量是许多常用的高序对流扩展的核心,包括按部对数对数对数的加勒金光谱元元元素方法(或光谱共位方法),即使这个办法不线性稳定,而且可能对模拟结果有不利影响。我们进一步从数字上表明,在模拟期间,这种计划会造成指数性两点保值的双点通通通量通量变化,因为它们可能分解或反分解。不幸的是,这些通量通量性通量的通量是许多常用的高序酶扩展的延伸扩展,例如平平平方方程式中的一种常见的平极式的平极式的平极式正方程式。