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A new, high-order slope-limiting procedure for the Piecewise Parabolic Method(PPM) and the Piecewise Quartic Method (PQM) is described. Following a WeightedEssentially Non-Oscillatory (WENO)-type paradigm, the proposed slope-limiterseeks to reconstruct smooth, non-oscillatory piecewise polynomial profiles as anon-linear combination of the natural and monotone-limited PPM and PQMinterpolants. Compared to existing monotone slope-limiting techniques, this newstrategy is designed to improve accuracy at smooth extrema, while controllingspurious oscillations in the neighbourhood of sharp features. Using the newslope-limited PPM and PQM interpolants, a high-order accurateArbitrary-Lagrangian-Eulerian framework for advection-dominated flows isconstructed, and its effectiveness is examined using a series of one- andtwo-dimensional benchmark cases. It is shown that the new WENO-typeslope-limiting techniques offer a significant improvement in accuracy comparedto existing strategies, allowing the PPM- and PQM- based schemes to achievefully third- and fifth-order accurate convergence, respectively, forsufficiently smooth problems.

A WENO-type slope-limiter for a family of piecewise polynomial methods