  In our previous studies [17, 18], the commonly reported issue that most of
the existing mapped WENO schemes suffer from either losing high resolutions or
generating spurious oscillations in long-run simulations of hyperbolic problems
has been successfully addressed, by devising the improved mapped WENO schemes,
namely MOP-WENO-X, where "X" stands for the version of the existing mapped WENO
scheme. However, all the MOP-WENO-X schemes bring about the serious deficiency
that their resolutions in the region with high-frequency but smooth waves are
dramatically decreased compared to their associated WENO-X schemes. The purpose
of this paper is to overcome this drawback. We firstly present the definition
of the locally order-preserving (LOP) mapping. Then, by using a new proposed
posteriori adaptive technique, we apply this LOP property to obtain the new
mappings from those of the WENO-X schemes. The essential idea of the posteriori
adaptive technique is to identify the global stencil in which the existing
mappings fail to preserve the LOP property, and then replace the mapped weights
with the weights of the classic WENO-JS scheme to recover the LOP property. We
build the resultant mapped WENO schemes and denote them as LOP-WENO-X. The
numerical experiments demonstrate that the resolutions in the region with
high-frequency but smooth waves of the LOP-WENO-X schemes are similar or even
better than those of their associated WENO-X schemes and naturally much higher
than the MOP-WENO-X schemes. Furthermore, the LOP-WENO-X schemes gain all the
great advantages of the MOP-WENO-X schemes, such as attaining high resolutions
and in the meantime preventing spurious oscillations near discontinuities when
solving the one-dimensional linear advection problems with long output times,
and significantly reducing the post-shock oscillations in the simulations of
the two-dimensional problems with shock waves.
