An improved CSPM approximation for multi-dimensional second-order derivatives

Many SPH approximations for second-order derivatives, or the Laplacian, suffer from the presence of boundaries and irregularities in the particle distribution. In this paper we discuss four estimates to the Laplacian: the Brookshaw approximation, CSPM, MSPH and ICSPM { the latter of which is derived in this work. We theoretically derive the convergence rate of these methods and validate the results with numerical experiments. We show that the widely used Brookshaw method suffers the most from boundaries or random particles, while MSPH and the method derived in this work (ICSPM) are able to stay accurate. ICSPM uses smaller matrices then MSPH and is therefore computationally more attractive.