Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic
  Turbulence Simulation
  Stable, accurate, divergence-free simulation of magnetized supersonic
turbulence is a severe test of numerical MHD schemes and has been surprisingly
difficult to achieve due to the range of flow conditions present. Here we
present a new, higher order-accurate, low dissipation numerical method which
requires no additional dissipation or local "fixes" for stable execution. We
describe PPML, a local stencil variant of the popular PPM algorithm for solving
the equations of compressible ideal magnetohydrodynamics. The principal
difference between PPML and PPM is that cell interface states are evolved
rather that reconstructed at every timestep, resulting in a compact stencil.
Interface states are evolved using Riemann invariants containing all transverse
derivative information. The conservation laws are updated in an unsplit
fashion, making the scheme fully multidimensional. Divergence-free evolution of
the magnetic field is maintained using the higher order-accurate constrained
transport technique of Gardiner and Stone. The accuracy and stability of the
scheme is documented against a bank of standard test problems drawn from the
literature. The method is applied to numerical simulation of supersonic MHD
turbulence, which is important for many problems in astrophysics, including
star formation in dark molecular clouds. PPML accurately reproduces in
three-dimensions a transition to turbulence in highly compressible isothermal
gas in a molecular cloud model. The low dissipation and wide spectral bandwidth
of this method make it an ideal candidate for direct turbulence simulations.
