The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

The choice of a time integration scheme is a crucial aspect of any transient fluid simulation, and Smoothed-Particle Hydrodynamics (SPH) is no exception. The influence of the time integration scheme on energy balance is here addressed. To do so, explicit expressions allowing to compute the deviations from the energy balance, induced by the time integration scheme, are provided. These expressions, computed $\mathit{\text{a}}\mathit{\text{posteriori}}$, are valid for different integration methods. Such formulation is tested with the simulation of a two-dimensional non-viscous impact of two jets, with no artificial dissipation terms. To the best of our knowledge, this is the first stable simulation of a non-dissipative system with a weakly-compressible SPH method. A viscous case, the Taylor-Green vortex, has also been simulated. Results show that an implicit time integration scheme also behaves better in a viscous context.