  Cosmological N-Body simulations are used for a variety of applications.
Indeed progress in the study of large scale structures and galaxy formation
would have been very limited without this tool. For nearly twenty years the
limitations imposed by computing power forced simulators to ignore some of the
basic requirements for modeling gravitational instability. One of the
limitations of most cosmological codes has been the use of a force softening
length that is much smaller than the typical inter-particle separation. This
leads to departures from collisionless evolution that is desired in these
simulations. We propose a particle based method with an adaptive resolution
where the force softening length is reduced in high density regions while
ensuring that it remains well above the local inter-particle separation. The
method, called the Adaptive TreePM, is based on the TreePM code. We present the
mathematical model and an implementation of this code, and demonstrate that the
results converge over a range of options for parameters introduced in
generalizing the code from the TreePM code. We explicitly demonstrate
collisionless evolution in collapse of an oblique plane wave. We compare the
code with the fixed resolution TreePM code and also an implementation that
mimics adaptive mesh refinement methods and comment on the agreement, and
disagreements in the results. We find that in most respects the ATreePM code
performs at least as well as the fixed resolution TreePM in highly over-dense
regions, from clustering and number density of haloes, to internal dynamics of
haloes. We also show that the adaptive code is faster than the corresponding
high resolution TreePM code.
