Techniques for second-order convergent weakly compressible smoothed particle hydrodynamics schemes without boundaries

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.