Standard formulations of smoothed particle hydrodynamics (SPH) are unable to resolve mixing at fluid boundaries. We use an error and stability analysis of the generalized SPH equations of motion to prove that this is due to two distinct problems. The first is a leading order error in the momentum equation. This should decrease with an increasing neighbour number, but does not because numerical instabilities cause the kernel to be irregularly sampled. We identify two important instabilities: the clumping instability and the banding instability, and we show that both are cured by a suitable choice of kernel. The second problem is the local mixing instability (LMI). This occurs as particles attempt to mix on the kernel scale, but are unable to due to entropy conservation. The result is a pressure discontinuity at boundaries that pushes fluids of different entropies apart. We cure the LMI by using a weighted density estimate that ensures that pressures are single-valued throughout the flow. This also gives a better volume estimate for the particles, reducing errors in the continuity and momentum equations. We demonstrate mixing in our new optimized smoothed particle hydrodynamics (OSPH) scheme using a Kelvin-Helmholtz instability (KHI) test with a density contrast of 1:2, and the 'blob test' -a 1:10 density ratio gas sphere in a wind tunnel -finding excellent agreement between OSPH and Eulerian codes.

We explore the initial conditions for fragments in the extended regions (r 50 AU) of gravitationally unstable disks. We combine analytic estimates for the fragmentation of spiral arms with 3D SPH simulations to show that initial fragment masses are in the gas giant regime. These initial fragments will have substantial angular momentum, and should form disks with radii of a few AU.We show that clumps will survive for multiple orbits before they undergo a second, rapid collapse due to H 2 dissociation and that it is possible to destroy bound clumps by transporting them into the inner disk. The consequences of disrupted clumps for planet formation, dust processing, and disk evolution are discussed. We argue that it is possible to produce Earth-mass cores in the outer disk during the earliest phases of disk evolution.

We present a novel implementation of smoothed particle hydrodynamics that uses the spatial derivative of the velocity divergence as a higher order dissipation switch. Our switch – which is second order accurate – detects flow convergence before it occurs. If particle trajectories are going to cross, we switch on the usual SPH artificial viscosity, as well as conservative dissipation in all advected fluid quantities (e.g. the entropy). The viscosity and dissipation terms (that are numerical errors) are designed to ensure that all fluid quantities remain single valued as particles approach one another, to respect conservation laws, and to vanish on a given physical scale as the resolution is increased. SPHS alleviates a number of known problems with ‘classic’ SPH, successfully resolving mixing, and recovering numerical convergence with increasing resolution. An additional key advantage is that – treating the particle mass similarly to the entropy – we are able to use multimass particles, giving significantly improved control over the refinement strategy. We present a wide range of code tests including the Sod shock tube, Sedov–Taylor blast wave, Kelvin–Helmholtz Instability, the ‘blob test’ and some convergence tests. Our method performs well on all tests, giving good agreement with analytic expectations.

The formation of the massive young stars surrounding SgrA * is still an open question. In this paper, we simulate the infall of a turbulent molecular cloud towards the Galactic Center (GC). We adopt two different cloud masses (4.3 × 10 4 M ⊙ and 1.3 × 10 5 M ⊙ ). We run five simulations: the gas is assumed to be isothermal in four runs, whereas radiative cooling is included in the fifth run. In all the simulations, the molecular cloud is tidally disrupted, spirals towards the GC, and forms a small, dense and eccentric disk around SgrA * . With high resolution simulations, we follow the fragmentation of the gaseous disk. Star candidates form in a ring at ∼ 0.1 − 0.4 pc from the super-massive black hole (SMBH) and have moderately eccentric orbits (e ∼ 0.2−0.4), in good agreement with the observations. The mass function of star candidates is top-heavy only if the local gas temperature is high ( > ∼ 100 K) during the star formation and if the parent cloud is sufficiently massive ( > ∼ 10 5 M ⊙ ). Thus, this study indicates that the infall of a massive molecular cloud is a viable scenario for the formation of massive stars around SgrA * , provided that the gas temperature is kept sufficiently high ( > ∼ 100 K).

We present 3D smoothed particle hydrodynamics simulations of the collapse of clumps formed through gravitational instability in the outer part of a protoplanetary disc. The initial conditions are taken directly from a global disc simulation, and a realistic equation of state is used to follow the clumps as they contract over several orders of magnitude in density, approaching the molecular hydrogen dissociation stage. The effects of clump rotation, asymmetries and radiative cooling are studied. Rotation provides support against fast collapse, but non‐axisymmetric modes develop and efficiently transport angular momentum outwards, forming a circumplanetary disc. This transport helps the clump reach the dynamical collapse phase, resulting from molecular hydrogen dissociation, on a thousand‐year time‐scale, which is smaller than time‐scales predicted by some previous spherical 1D collapse models. Extrapolation to the threshold of the runaway hydrogen dissociation indicates that the collapse time‐scales can be shorter than inward migration time‐scales, suggesting that clumps could survive tidal disruption and deliver a protogas giant to distances of even a few au from the central star.