  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.
