A new class of Moving-Least-Squares WENO–SPH schemes

Abstract

We present a new class of meshless Lagrangian particle methods based on the SPH formulation of Vila and Ben Moussa, combined with a new weighted essentially non-oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The key idea is to produce for each particle first a set of high order accurate Moving-Least-Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a non-linear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS–WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non-oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes.To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions.We test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes.