Cosmological N-Body simulations are used for a variety of applications.Indeed progress in the study of large scale structures and galaxy formationwould have been very limited without this tool. For nearly twenty years thelimitations imposed by computing power forced simulators to ignore some of thebasic requirements for modeling gravitational instability. One of thelimitations of most cosmological codes has been the use of a force softeninglength that is much smaller than the typical inter-particle separation. Thisleads to departures from collisionless evolution that is desired in thesesimulations. We propose a particle based method with an adaptive resolutionwhere the force softening length is reduced in high density regions whileensuring that it remains well above the local inter-particle separation. Themethod, called the Adaptive TreePM, is based on the TreePM code. We present themathematical model and an implementation of this code, and demonstrate that theresults converge over a range of options for parameters introduced ingeneralizing the code from the TreePM code. We explicitly demonstratecollisionless evolution in collapse of an oblique plane wave. We compare thecode with the fixed resolution TreePM code and also an implementation thatmimics adaptive mesh refinement methods and comment on the agreement, anddisagreements in the results. We find that in most respects the ATreePM codeperforms at least as well as the fixed resolution TreePM in highly over-denseregions, from clustering and number density of haloes, to internal dynamics ofhaloes. We also show that the adaptive code is faster than the correspondinghigh resolution TreePM code.
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