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

TL;DR

This paper systematically studies the convergence issues in weakly-compressible SPH schemes and proposes new second-order accurate variations that improve accuracy and conservation properties.

Contribution

It identifies error sources preventing second-order convergence and introduces new WCSPH schemes with second-order accuracy and better conservation.

Findings

New WCSPH schemes achieve second-order convergence.

Proper kernel and discretization choices improve accuracy.

Schemes demonstrate improved momentum conservation.

Abstract

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.