In this thesis we study Smoothed Particle Hydrodynamics (SPH) which is a numerical method for simulating fluid flow used widely in astrophysics. In SPH artificial viscosity is necessary for the correct treatment of shocks, but often generates unwanted dissipation away from shocks, particularly in poorly resolved flows. In this study we address this problem by refining the method proposed by Morris & Monaghan (1997). The new scheme uses the rate of change of the velocity divergence, Dt(∇•v), to indicate a shock and focuses on eliminating viscosity away from shocks. The new method works as least as well as any previous technique in the strong-shock regime, but becomes virtually inviscid away from shocks. In particular sound waves or oscillations of self-gravitating gas spheres are hardly damped over many periods. \ud We also look at stability issues for SPH, in particular the well known clumping instability. We perform numerical tests of the stability analysis performed by Morris (1996) and find that there are bands of unstable regions as suggested by Read et al. (2010). We also demonstrate that a cored kernel can greatly reduce the clumping instability.\ud Finally we apply the SPH method to extend the stellar disruption work of Lodato et al. (2009) to orbits with a range of pericentre distances. We find that the light curve produced by this disruption is closer to the predicted L ∝ t−5/3 (Rees, 1988) for encounters that are closer to the black hole than the tidal disruption radius. For encounters that are further from the black hole than the tidal disruption radius, the light curve deviates from this predicted power law. We also look at how elliptical orbits can effect the stability of the star. We find that in elliptical orbits a star can be disturbed further from the black hole than in the parabolic case. We then consider the fate of the S2 star and conclude that when it becomes a red giant and expands, S2 will be tidally disrupted
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