  A new, high-order slope-limiting procedure for the Piecewise Parabolic Method
(PPM) and the Piecewise Quartic Method (PQM) is described. Following a Weighted
Essentially Non-Oscillatory (WENO)-type paradigm, the proposed slope-limiter
seeks to reconstruct smooth, non-oscillatory piecewise polynomial profiles as a
non-linear combination of the natural and monotone-limited PPM and PQM
interpolants. Compared to existing monotone slope-limiting techniques, this new
strategy is designed to improve accuracy at smooth extrema, while controlling
spurious oscillations in the neighbourhood of sharp features. Using the new
slope-limited PPM and PQM interpolants, a high-order accurate
Arbitrary-Lagrangian-Eulerian framework for advection-dominated flows is
constructed, and its effectiveness is examined using a series of one- and
two-dimensional benchmark cases. It is shown that the new WENO-type
slope-limiting techniques offer a significant improvement in accuracy compared
to existing strategies, allowing the PPM- and PQM- based schemes to achieve
fully third- and fifth-order accurate convergence, respectively, for
sufficiently smooth problems.
