Accurate, non-oscillatory, remeshing schemes for particle methods

In this article we propose and validate new remeshing schemes for the simulation of transport equations by particle methods. Particle remeshing is a common way to control the regularity of the particle distribution which is necessary to guarantee the accuracy of particle methods in presence of strong strain in a flow. Using a grid-based analysis, we derive remeshing schemes that can be used in a consistent way at every time-step in a particle method. The schemes are obtained by local corrections of classical third order and fifth order interpolation kernels. The time-step to be used in the resulting push-and-remesh particle method is determined on the basis of rigorous bounds and can significantly exceed values obtained by CFL conditions in usual grid-based Eulerian methods. In addition, we extend the analysis of [5] to obtain TVD remeshing schemes that avoid oscillations of remeshing formulas near sharp variations of the solution. These methods are illustrated in several flow conditions in 1D, 2D and 3D.