Energy conservation in the -SPH scheme

An in-depth analysis of the energy conservation in the ?-SPH model has been carried on. In comparison to the standard SPH scheme, the mechanical energy equation of the ?-SPH variant is characterized by a further term that is generally dissipative and is related to the diffusive operator inside the continuity equation. The behaviour and the structure of such a term have been studied in detail and a number of specifically conceived test cases has been considered, highlighting that the dissipative term is generally small and it mainly acts when spurious high-frequency acoustic components are excited.