SPH truncation error in estimating a 3D function

The SPH truncation error (εT) 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 εT 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 εT has been performed adopting regular and irregular distributions of particles, arranged within a cube, delimited by lateral walls at each side. The evolution of εT 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 εT 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.