Simulating the Complexity of the Dark Matter Sheet I: Numerical
  Algorithms
  At early times dark matter has a thermal velocity dispersion of unknown
amplitude which, for warm dark matter models, can influence the formation of
nonlinear structure on observable scales. We propose a new scheme to simulate
cosmologies with a small-scale suppression of perturbations that combines two
previous methods in a way that avoids the numerical artefacts which have so far
prevented either from producing fully reliable results. At low densities and
throughout most of the cosmological volume, we represent the dark matter
phase-sheet directly using high-accuracy interpolation, thereby avoiding the
artificial fragmentation which afflicts particle-based methods in this regime.
Such phase-sheet methods are, however, unable to follow the rapidly increasing
complexity of the denser regions of dark matter haloes, so for these we switch
to an N-body scheme which uses the geodesic deviation equation to track
phase-sheet properties local to each particle. In addition, we present a novel
high-resolution force calculation scheme based on an oct-tree of cubic force
resolution elements which is well suited to approximate the force-field of our
combined sheet+particle distribution. Our hybrid simulation scheme enables the
first reliable simulations of the internal structure of low-mass haloes in a
warm dark matter cosmology.
