Adaptive mesh refinement method. Part 1: Automatic thresholding based on a distribution function

The accurate numerical simulation of large scale flows, together with the detailed modeling of flooding or drying of small-scale regions, is a difficult and a challenging problem. Adaptive mesh method allows, in principle, to solve accurately those scales. However in practice, on one hand, the lack of a priori or efficient a posteriori error estimates, especially for multidimensional hyperbolic problems, make the analysis harder. On the other hand, once a mesh refinement criterion is chosen, the difficult problem is to determine the mesh refinement threshold parameter which is certainly the most important part of the adaptive process. The smaller this parameter is, the higher the number of cells refined is at the expense of the computational cost. In this work, we numerically investigate different refinement criteria and we present a general procedure to determine automatically a mesh refinement threshold for any given mesh refinement criterion. To this end the decreasing rearrangement (distribution) function of the mesh refinement criterion is introduced to catch relevant scales. The efficiency of the automatic thresholding method is illustrated through the one dimensional Saint-Venant system. Multidimensional and real life applications such as Tsunamis propagations are dealt in the second part