  Numerical methods to improve the treatment of magnetic fields in smoothed
field magnetohydrodynamics (SPMHD) are developed and tested. Chapter 2 is a
review of SPMHD. In Chapter 3, a mixed hyperbolic/parabolic scheme is developed
which cleans divergence error from the magnetic field. Average divergence error
is an order of magnitude lower for all test cases considered, and allows for
the stable simulation of the gravitational collapse of magnetised molecular
cloud cores. The effectiveness of the cleaning may be improved by explicitly
increasing the hyperbolic wave speed or by cycling the cleaning equations
between timesteps. In the latter, it is possible to achieve DivB=0. Chapter 4
develops a switch to reduce dissipation of the magnetic field from artificial
resistivity. Compared to the existing switch in the literature, this leads to
sharper shock profiles in shocktube tests, lower overall dissipation of
magnetic energy, and importantly, is able to capture magnetic shocks in the
highly super-Alfvenic regime. Chapter 5 compares these numerical methods
against grid-based MHD methods (using the Flash code) in simulations of the
small-scale dynamo amplification of a magnetic field in driven, isothermal,
supersonic turbulence. Both codes exponentially amplify the magnetic energy at
a constant rate, though SPMHD shows a resolution dependence that arises from
the scaling of the numerical dissipation terms. The time-averaged saturated
magnetic spectra have similar shape, and both codes have PDFs of magnetic field
strength that are log-normal, which become lopsided as the magnetic field
saturates. We conclude that SPMHD is able to reliably simulate the small-scale
dynamo amplification of magnetic fields. Chapter 6 concludes the thesis and
presents some preliminary work demonstrating that SPMHD can activate the
magneto-rotational instability in 2D shearing box tests.
