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Adaptive mesh refinement strategy for a non conservative transport problem

Long time simulations of transport equations raise computational challenges since they require both a large domain of calculation and sufficient accuracy. It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small cells in the region of interest and coarser cells where the solution is smooth. Biological models involving cell dynamics fall for instance within this framework and are often non conservative to account for cell division. In that case the threshold controlling the spatial adaptivity may have to be time-dependent in order to keep up with the progression of the solution. In this article we tackle the difficulties arising when applying a Multiresolution method to a transport equation with discontinuous fluxes modeling localized mitosis. The analysis of the numerical method is performed on a simplified model and numerical scheme. An original threshold strategy is proposed and validated thanks to extensive numerical tests. It is then applied to a biological model in both cases of distributed and localized mitosis.