SPH truncation error in estimating a 3D derivative

Following the procedure proposed in Quinlan et al. (Int. J. Numer. Meth. Engng. 2006; 66: 2064-2085) for a 1D generic derivative, a 3D formulation of the Smoothed Particle Hydrodynamics (SPH) truncation error (epsilon(T)) has been derived and validated. We have then underlined the differences between traditional SPH simulations, which are not consistent, and estimations using renormalization, a first-order consistency technique. The consistency order is here defined as the highest degree of a generic polynomial function, which can be exactly reproduced by an SPH approximation. Under the homogeneous conditions assumed in our analyses renormalization generally reduces the relative truncation error by 1 or 2 orders of magnitude, both at inner points and boundary locations. Due to renormalization the error tends to a lowest constant value as the kernel support size (h) goes to zero, while in general with no consistency the error behaves like 1/h. In contrast to formulations without any consistency estimations, using renormalization there is a weak dependence of the error on the absolute value of the displacement of the particles from their volume barycentre (delta). In addition, for simulations with renormalization, the best choice for the kernel function seems to be the closest to Dirac's delta, while for the ones with no consistency, the preferences are altered. Furthermore, we observe that renormalization reduces the number of neighbors that are necessary to obtain a discretization error that is negligible with respect to the integral error. Copyright (C) 2011 John Wiley & Sons, Ltd.