  In this paper we propose an accurate, and computationally efficient method
for incorporating adaptive spatial resolution into weakly-compressible Smoothed
Particle Hydrodynamics (SPH) schemes. Particles are adaptively split and merged
in an accurate manner. Critically, the method ensures that the number of
neighbors of each particle is optimal, leading to an efficient algorithm. A set
of background particles is used to specify either geometry-based spatial
resolution, where the resolution is a function of distance to a solid body, or
solution-based adaptive resolution, where the resolution is a function of the
computed solution. This allows us to simulate problems using particles having
length variations of the order of 1:250 with much fewer particles than
currently reported with other techniques. The method is designed to
automatically adapt when any solid bodies move. The algorithms employed are
fully parallel. We consider a suite of benchmark problems to demonstrate the
accuracy of the approach. We then consider the classic problem of the flow past
a circular cylinder at a range of Reynolds numbers and show that the proposed
method produces accurate results with a significantly reduced number of
particles. We provide an open source implementation and a fully reproducible
manuscript.
