An approach for solving the boundary free edge difficulties in SPH modelling: application to a viscous accretion disc in close binaries

Abstract

In this work, we propose a SPH interpolating Kernel reformulation suitable also to treat free edge boundaries in the computational domain. Application to both inviscid and viscous stationary low compressibility accretion disc models in Close Binaries (CB) are shown. The investigation carried out in this paper is a consequence of the fact that a low compressibility modelling is crucial to check numerical reliability. Results show that physical viscosity supports a well-bound accretion disc formation, despite the low gas compressibility, when a Gaussian-derived Kernel (from the Error Function) is assumed, in extended particle range - whose Half Width at Half Maximum (HWHM) is fixed to a constant $h$ value - without any spatial restrictions on its radial interaction (hereinafter GASPHER). At the same time, GASPHER ensures adequate particle interpolations at the boundary free edges. Both SPH and adaptive SPH (hereinafter ASPH) methods lack accuracy if there are not constraints on the boundary conditions, in particular at the edge of the particle envelope: Free Edge (FE) conditions. In SPH, an inefficient particle interpolation involves a few neighbour particles; instead, in the second case, non-physical effects involve both the boundary layer particles themselves and the radial transport. Either in a regime where FE conditions involve the computational domain, or in a viscous fluid dynamics, or both, a GASPHER scheme can be rightly adopted in such troublesome physical regimes. Despite the applied low compressibiity condition, viscous GASPHER model shows clear spiral pattern profiles demonstrating the better quality of results compared to SPH viscous ones. Moreover a successful comparison of results concerning GASPHER 1D inviscid shock tube with analytical solution is also reported.