Analysis through energy consideration of a quasi-Lagrangian scheme using Riemann stabilization

The Smoothed Particle Hydrodynamics (SPH) method suffers from the presence of anisotropic particle configurations inherent to its Lagrangian nature. A way to circumvent this problem is represented by the Particle Shifting Techniques (PSTs). As recently shown, a PST can be included in the SPH schemes through an Arbitrary-Lagrangian-Eulerian (ALE) formalism, or variants of it. This choice becomes relevant when the problems under investigation involve confined flows or violent sloshing motions. In the present paper, we introduce an SPH scheme with PST terms similar to those described in [Antuono et al. CaF, 2021], but using Riemann stabilization technique. The energy balance of this scheme is then analysed in detail. In particular, the energy dissipated by the Riemann terms is studied following the same approach used for the ?-SPH model. Differently from that work, however, additional terms (mainly related to the use of the PST) appear in the energy balance. A specific attention is then paid to the diffusive and consistency aspects of these energy terms, on different test cases.