The small-scale turbulent dynamo in smoothed particle
  magnetohydrodynamics
  Supersonic turbulence is believed to be at the heart of star formation. We
have performed smoothed particle magnetohydrodynamics (SPMHD) simulations of
the small-scale dynamo amplification of magnetic fields in supersonic
turbulence. The calculations use isothermal gas driven at rms velocity of Mach
10 so that conditions are representative of star-forming molecular clouds in
the Milky Way. The growth of magnetic energy is followed for 10 orders in
magnitude until it reaches saturation, a few percent of the kinetic energy. The
results of our dynamo calculations are compared with results from grid-based
methods, finding excellent agreement on their statistics and their qualitative
behaviour. The simulations utilise the latest algorithmic developments we have
developed, in particular, a new divergence cleaning approach to maintain the
solenoidal constraint on the magnetic field and a method to reduce the
numerical dissipation of the magnetic shock capturing scheme. We demonstrate
that our divergence cleaning method may be used to achieve $\nabla \cdot {\bf
B}=0$ to machine precision, albeit at significant computational expense.
