  We study nonlinear hyperbolic conservation laws with non-convex flux in one
space dimension and, for a broad class of numerical methods based on summation
by parts operators, we compute numerically the kinetic functions associated
with each scheme. As established by LeFloch and collaborators, kinetic
functions (for continuous or discrete models) uniquely characterize the
macro-scale dynamics of small-scale dependent, undercompressive, nonclassical
shock waves. We show here that various entropy-dissipative numerical schemes
can yield nonclassical solutions containing classical shocks, including Fourier
methods with (super-) spectral viscosity, finite difference schemes with
artificial dissipation, discontinuous Galerkin schemes with or without modal
filtering, and TeCNO schemes. We demonstrate numerically that entropy stability
does not imply uniqueness of the limiting numerical solutions for scalar
conservation laws in one space dimension, and we compute the associated kinetic
functions in order to distinguish between these schemes. In addition, we design
entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are
measures with delta shocks. This system illustrates the fact that entropy
stability does not imply boundedness under grid refinement.
