Implicit integrations for SPH in semi-Lagrangian approach: Application to the accretion disc modeling in a microquasar

Current explicit integration techniques in fluid dynamics are deeply limited by the Courant-Friedrichs-Lewy condition of the time step progression, based on the adopted spatial resolution coupled with the maximum value between the kinetic velocity or the signal transmission speed in the computational domain. Eulerian implicit integration techniques, even though more time consuming, can allow us to perform stable computational fluid dynamics paying the price of a relatively larger inaccuracy in the calculations, without suffering such a strict temporal limitation. In this paper, we present a simple and effective scheme to perform free Lagrangian Smooth Particle Hydrodynamics (SPH) implicit integrations in the semi-Lagrangian approach without any Jacobian matrix inversion operations for viscous Navier-Stokes flows. Applications to SPH accretion disc simulation around a massive black hole (MBH) in a binary stellar system are shown, together with the comparison to the same results obtained according to the traditional explicit integration techniques. Some 1D and 2D critical tests are also discussed to check the validity of the technique.