A way to increase the convergence-order in SPH

Smoothed Particle Hydrodynamics (SPH) schemes generally result from a compromise between the need for conservation of the main global quantities ( global mass, linear and angular momenta) and consistency of the SPH mollified operators towards the exact ones. In the present work, a globally non-conservative but locally accurate pressure gradient approximation is adopted, resulting in a novel high-order weakly-compressible SPH scheme, which also prevents Tensile Instability occurrences. To fulfill the dynamic free-surface boundary condition, a switch between the non-conservative and conservative formulations is retained, the latter being preferred in a thin region close to the free-surface. Regular particle distributions are maintained thanks to recent improvements in Particle Shifting Techniques. The latter are taken into account within the continuity and momentum equations through a quasi-Lagrangian formalism. The numerical diffusion is obtained using Riemann solvers, with a reconstruction/limitation of the left and right states based on the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) technique, allowing for low numerical dissipation without tuning parameters. The numerical investigation is carried out on several problems characterized by different flow features and showing the advantages of the present scheme with respect to conservative formulations. Since the proposed formulation does not intrinsically guarantee momenta conservation, the latter are monitored proving that the overall errors are generally acceptable.