A third-order Energy Stable  Weighted
Essentially Non--Oscillatory (ESWENO) finite difference
scheme developed by the authors of the paper [N. K. Yamaleev and
M. H. Carpenter, Third-order energy stable WENO scheme,  (J. of
Comput. Phys.), was proven to be stable in the energy norm for both
continuous and discontinuous solutions of systems of linear hyperbolic
equations.  Herein, a systematic approach is presented that enables
``energy stable'' modifications for existing WENO schemes of any
order.  The technique is demonstrated by developing a one-parameter
family of fifth-order upwind-biased ESWENO schemes including one
sixth-order central scheme; ESWENO schemes up to eighth order are also
available. We also develop new weight functions and derive constraints
on their parameters, which provide consistency, much faster
convergence of the high-order ESWENO schemes to their underlying
linear schemes for smooth solutions with arbitrary number of vanishing
derivatives, and better resolution near strong discontinuities than
the conventional WENO counterparts.