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We introduce a novel location-based moving mesh algorithm MMSISL in which the arrival points in the Semi-Implicit Semi-Lagrangian (SISL) algorithm are located by using an equidistribution strategy. This algorithm gives a natural coupling between moving mesh methods and SISL methods. It involves little extra cost in implementation as it exploits the interpolation methods already embedded in the SISL algorithm. We apply this method to a number of partial differential equation problems in one-dimension, each of which have sharply defined features. We show that using MMSISL leads to a markedly improved performance over fixed mesh methods, with significantly reduced errors. We also show that unlike many adaptive schemes, no issues arise in the MMSISL algorithm from a CFL condition imposed restriction on the time step.