  Existing mapped WENO schemes can hardly prevent spurious oscillations while
preserving high resolutions at long output times. We reveal in this paper the
essential reason for such phenomena. It is actually caused by that the mapping
function in these schemes can not preserve the order of the nonlinear weights
of the stencils. The nonlinear weights may be increased for non-smooth stencils
and be decreased for smooth stencils. It is then indicated to require the set
of mapping functions to be Order-Preserving in mapped WENO schemes. Therefore,
we propose a new mapped WENO scheme with a set of mapping functions to be
order-preserving which exhibits a remarkable advantage over the mapped WENO
schemes in references. For long output time simulations, the new scheme has the
capacity to attain high resolutions and avoid spurious oscillations near
discontinuities meanwhile.
