  Despite the many advances in the use of weakly-compressible smoothed particle
hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is
still challenging to obtain second-order convergence even for simple periodic
domains. In this paper we perform a systematic numerical study of convergence
and accuracy of kernel-based approximation, discretization operators, and
weakly-compressible SPH (WCSPH) schemes. We explore the origins of the errors
and issues preventing second-order convergence despite having a periodic
domain. Based on the study, we propose several new variations of the basic
WCSPH scheme that are all second-order accurate. Additionally, we investigate
the linear and angular momentum conservation property of the WCSPH schemes. Our
results show that one may construct accurate WCSPH schemes that demonstrate
second-order convergence through a judicious choice of kernel, smoothing
length, and discretization operators in the discretization of the governing
equations.
