An adaptively refined phase-space element method for cosmological
  simulations and collisionless dynamics
  N-body simulations are essential for understanding the formation and
evolution of structure in the Universe. However, the discrete nature of these
simulations affects their accuracy when modelling collisionless systems. We
introduce a new approach to simulate the gravitational evolution of cold
collisionless fluids by solving the Vlasov-Poisson equations in terms of
adaptively refineable "Lagrangian phase space elements". These geometrical
elements are piecewise smooth maps between Lagrangian space and Eulerian phase
space and approximate the continuum structure of the distribution function.
They allow for dynamical adaptive splitting to accurately follow the evolution
even in regions of very strong mixing. We discuss in detail various one-, two-
and three-dimensional test problems to demonstrate the performance of our
method. Its advantages compared to N-body algorithms are: i) explicit tracking
of the fine-grained distribution function, ii) natural representation of
caustics, iii) intrinsically smooth gravitational potential fields, thus iv)
eliminating the need for any type of ad-hoc force softening. We show the
potential of our method by simulating structure formation in a warm dark matter
scenario. We discuss how spurious collisionality and large-scale discreteness
noise of N-body methods are both strongly suppressed, which eliminates the
artificial fragmentation of filaments. Therefore, we argue that our new
approach improves on the N-body method when simulating self-gravitating cold
and collisionless fluids, and is the first method that allows to explicitly
follow the fine-grained evolution in six-dimensional phase space.
