On the construction of mass conservative and meshless adaptive particle advection methods

This contribution concerns the construction of meshless Lagrangian particle methods for the numerical simulation of multiscale phenomena in linear transport problems, where mass conservative discretization methods are essentially required. The proposed discretization scheme works with a finite set of unstructured nodes, each corresponding to one flow particle. In this method, corresponding particle average values are maintained during the simulation. The discrete nodes are subject to adaptive modifications, leading to semi-Lagrangian particle simulations, whose adaption rules rely on the insertion (refinement) and removal (coarsening) of the nodes at each time step. The resulting meshless particle method is mass conservative by construction. The required algebraic rules for the downstream particle advection and the local redistribution of the particle masses are developed. Moreover, the implementation of boundary conditions is addressed. The efficacy of the proposed conservative and meshless adaptive particle method is finally shown by using one numerical simulation concerning the slotted cylinder, a popular standard test case for passive advection.