  For simulations that deal only with dark matter or stellar systems, the
conventional N-body technique is fast, memory efficient, and relatively simple
to implement. However when including the effects of gas physics, mesh codes are
at a distinct disadvantage compared to SPH. Whilst implementing the N-body
approach into SPH codes is fairly trivial, the particle-mesh technique used in
mesh codes to couple collisionless stars and dark matter to the gas on the
mesh, has a series of significant scientific and technical limitations. These
include spurious entropy generation resulting from discreteness effects, poor
load balancing and increased communication overhead which spoil the excellent
scaling in massively parallel grid codes.
  We propose the use of the collisionless Boltzmann moment equations as a means
to model collisionless material as a fluid on the mesh, implementing it into
the massively parallel FLASH AMR code. This approach, which we term
"collisionless stellar hydrodynamics" enables us to do away with the
particle-mesh approach. Since the parallelisation scheme is identical to that
used for the hydrodynamics, it preserves the excellent scaling of the FLASH
code already demonstrated on peta-flop machines.
  We find the classic hydrodynamic equations and Boltzmann moment equations can
be reconciled under specific conditions, allowing us to generate analytic
solutions for collisionless systems using conventional test problems. We
confirm the validity of our approach using a suite of demanding test problems,
including the use of a modified Sod shock test. We conclude by demonstrating
the ability of our code to model complex phenomena by simulating the evolution
of a spiral galaxy whose properties agree with those predicted by swing
amplification theory. (Abridged)
