A New Method for Approximating of First Derivatives in Smoothed Particle Hydrodynamics: Theory and Practice for Linear Transport Equation

Smoothed Particle Hydrodynamics (SPH) can be considered as a method for approximating dynamical partial differential equations on moving irregularly spaced nodes. The main idea of SPH is to construct a smooth approximating function for a discrete set of nodes using finite smoothing kernels whose support is smaller than the computational domain but covers several neighboring particles. In the classical version of SPH, the differentiation operation is applied to smooth kernels to calculate derivatives. In this paper, a new method for approximating first derivatives in SPH is proposed, based on the use of the idea of finite differences instead of kernel’s differentiating. It is shown that with comparable computational costs, the new method for calculating derivatives gives the same or higher order of approximation as the classical one. In addition, it was found that the actual error of the solution obtained by the new method, even with a rough resolution, is several times smaller than when using the classical method. This result was obtained theoretically by means of dispersion analysis and confirmed in practice when solving a linear one-dimensional transport equation.