  In this article, we propose a modified convex combination of the polynomial
reconstructions of odd-order WENO schemes to maintain the central substencil
prevalence over the lateral ones in all parts of the solution. New "centered"
versions of the classical WENO-Z and its less dissipative counterpart, WENO-Z+,
are defined through very simple modifications of the classical nonlinear
weights and show significantly superior numerical properties; for instance, a
well-known dispersion error for long-term runs is fixed, along with decreased
dissipation and better shock-capturing abilities. Moreover, the proposed
centered version of WENO-Z+ has no ad-hoc parameters and no dependence on the
powers of the grid size. All the new schemes are thoroughly analyzed concerning
convergence at critical points, adding to the discussion on the relevance of
such convergence to the numerical simulation of typical hyperbolic conservation
laws problems. Nonlinear spectral analysis confirms the enhancement achieved by
the new schemes over the standard ones.
