  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 \textit{a 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.
