High Order Semi-Lagrangian Particle Methods

Semi-lagrangian (or remeshed) particle methods are conservative particle methods where the particles are remeshed at each time-step. The numerical analysis of these methods show that their accuracy is governed by the regularity and moment properties of the remeshing kernel and that their stability is guaranteed by a lagrangian condition which does not rely on the grid size. Turbulent transport and more generally advection dominated flows are applications where these features make them appealing tools. The adaptivity of the method and its ability to capture fine scales at minimal cost can be further reinforced by remeshing particles on adapted grids, in particular through wavelet-based multi-resolution analysis. 1 Accuracy issues in particle methods Particle methods are not in general associated with the concept of high accuracy. They are instead viewed as numerical models, able to reproduce qualitative features of advection dominated phenomena even with few particles, in particular in situations with strongly unsteady dynamics. Examples of early applications of particle methods which illustrate these capabilities in flow simulations are transition to turbulence in wall bounded flows [23] or the study of vortex reconnection [25]. Free surface or compressible flows are other examples where particle methods can give an intuitive qualitative understanding of the flow dynamics in situations where Direct Numerical Simulations with classical discretization methods would require prohibitive computational resources. This is in particular true for applications in computer graphics [17] or in astrophysics [12, 20]. The numerical analysis of particle methods allows to understand the accuracy issues that these methods face. Particle methods are based on the concept that Dirac Georges-Henri Cottet University Grenoble Alpes and Institut Universitaire de France e-mail: [email protected]