Energy balance in the δδ-SPH scheme

• The theoretical analysis of the energy balance in the δδ-SPH model has been carried on.
• Inviscid free-surface problems have been used to study the model dissipation.
• δδ-SPH numerical dissipation mainly acts on spurious high-frequency acoustic components.
• Higher accuracy of the solutions with respect to the standard SPH model is observed.
• δδ-SPH prevents numerical accumulation from mechanical energy to compressible energy.

An in-depth analysis of the energy balance in the δδ-SPH model has been carried on. In comparison to the standard SPH scheme, the mechanical energy equation of the δδ-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 have been considered, highlighting that the dissipative term is generally small and it mainly acts when spurious high-frequency acoustic components are excited. In spite of such a dissipation mechanism, the δδ-SPH appears more accurate than the standard SPH scheme even in simulating inviscid fluids.