  The smoothed particle hydrodynamics (SPH) method has been widely used to
simulate incompressible and slightly compressible fluid flows. Adaptive
refinement strategies to dynamically increase the resolution of the particles
to capture sharp gradients in the flow have also been developed. However, most
of the SPH schemes in the literature are not second-order convergent (SOC).
Both second-order convergence and adaptive resolution are considered grand
challenge problems in the SPH community. In this paper, we propose, for the
first time, a second-order convergent (SOC) adaptive refinement strategy along
with a SOC weakly-compressible SPH scheme. We employ the method of manufactured
solutions to systematically develop the scheme and validate the solver. We
demonstrate the order of convergence of the entire scheme using the
Taylor-Green vortex problem and then go on to demonstrate the applicability of
the method to simulate flow past a circular cylinder.
