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

Abstract

Adaptive spatial domains are currently used in smooth particle hydrodynamics (SPH) with the aim of performing better spatial interpolations, mainly for expanding or shock gas dynamics. In this work, we propose an SPH interpolating kernel reformulation, also suitable for treating free edge (FE) boundaries in the computational domain. Application to both inviscid and viscous stationary low compressibility accretion disc models in close binaries is shown. The investigation carried out in this paper is a consequence of the fact that low compressibility modelling is crucial for checking numerical reliability. Results show that physical viscosity supports well-bound accretion disc formation, despite the low gas compressibility, when a Gaussian-derived kernel (from the error function) is assumed, in an extended particle rangewhose half-width at half-maximum is fixed to a constant h value -without any spatial restrictions on its radial interaction: Gaussian SPH in extended range (hereinafter GASPHER). At the same time, GASPHER ensures adequate particle interpolations at the boundary FEs. Both SPH and adaptive SPH (ASPH) methods lack accuracy if there are no constraints on the boundary conditions, in particular at the edge of the particle envelope: FE conditions. In SPH, an inefficient particle interpolation involves a few neighbour particles; however, in ASPH, non-physical effects involve both the boundary layer particles and the radial transport.A GASPHER scheme can be rightly adopted in troublesome physical regimes such as in a regime where FE conditions involve the computational domain, in viscous fluid dynamics or in both.Despite the low compressibiity condition applied, the viscous GASPHER model shows clear spiral pattern profiles demonstrating better quality of results compared to SPH viscous ones. Moreover, a successful comparison of results concerning a GASPHER 1D inviscid shock tube with an analytical solution is also reported.

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

“…On the contrary, spiral structures around MBH, as well as spiral shocks in the disc's radial flow, are normally seen in the 2D non viscous low compressibility modelling (Lanzafame & Belvedere 2005). In the case of low mass close binaries, SPH low compressibility viscous disc models are not able to show any spiral pattern because of several shortcomings in the method, especially when free edge conditions are considered (Lanzafame 2010b).…”

“…Even though a variable smoothing resolution length could also be used in SPH codes, the inadequacy of this approach in particular as for the radial transport whenever free boundary edges characterize the flow, recently discussed (Lanzafame 2010b;Lanzafame et al 2011), suggests us in using a constant h. However, the adoption of an implicit integration scheme to SPH is not correlated to its resolution length variability or constancy.…”