  A general dispersion-dissipation condition for finite difference schemes is
derived by analyzing the numerical dispersion and dissipation of explicit
finite-difference schemes. The proper dissipation required to damp spurious
high-wavenumber waves in the solution is determined from a physically motivated
relation between group velocity and dissipation rate. The application to a
previously developed low-dissipation weighted essentially non-oscillatory
scheme (WENO-CU6-M2) [X. Y. Hu and N. A. Adams; Scale separation for implicit
large eddy simulation, J. Comput. Phys. 230 (2011) 7240-7249] demonstrates that
this condition can serve as general guideline for optimizing the dispersion and
dissipation of linear and non-linear finite-difference schemes. Moreover, the
improved WENO-CU6-M2 which satisfies the dispersion-dissipation condition can
be used for under-resolved simulations. We demonstrate this capability by
considering transition to turbulence and self-similar energy decay of the
three-dimensional Taylor-Green vortex. Simulations of the inviscid and the
viscous Taylor-Green vortex at Reynolds numbers ranging from $Re=400$ to
$Re=3000$ show a significant improvement over the classical dynamic Smagorinsky
model and demonstrate competitiveness with state-of-the-art implicit LES
models, while preserving shock-capturing properties.
