SPH truncation error in estimating a 3D function

The SPH truncation error ( ε T ) can be defined as the sum of the integral kernel and the particle approximation error in Smoothed Particle Hydrodynamics modelling. Following the procedure proposed by Quinlan et al. [16] for a 1D generic derivative, we have derived an approximated 3D formulation of ε T in reproducing a generic function. This kind of estimation is implemented in some SPH models in order to reproduce density or some transported scalars. Then a corresponding sensitivity analysis of ε T has been performed adopting regular and irregular distributions of particles, arranged within a cube, delimited by lateral walls at each side. The evolution of ε T has been analyzed and compared to the proposed formulation, which has been numerically estimated under some simple conditions. The SPH truncation error has then been investigated on a simple free surface test case: a supercritical flow over a channel sill. We have developed some conclusions about the dependence of ε T on the position of the particles (inner or boundary), the shape of the function to be reproduced ( f), the kernel support size ( h), the particle volumes ( ω), the kernel function ( W), a non-dimensional distance between the volume barycentre and the particle location ( δ ), and a geometric anisotropy index of the particle volumes ( I ). We have finally underlined the difference between non-consistent simulations and estimations using Shepard’s correction.