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In this paper we focus on two sources of enhancement in accuracy and computational demanding in approximating a function and its derivatives by means of the Smoothed Particle Hydrodynamics method. The approximating power of the standard method is perceived to be poor and improvements can be gained making use of the Taylor expansion of the kernel approximation of the function and its derivatives. The modified formulation is appealing providing more accurate results of the function and its derivatives simultaneously without changing the kernel function adopted in the computation. The scheme received attention from practitioners, but many fundamental issues are still widely open. In this paper we highlight numerical insights of the scheme: studies on the accuracy, the convergence rate and the computational e↵orts with various data sites are provided. Accuracy of arbitrary order can be reached by employing the derivatives of the kernel with order up to the desired accuracy in approximating the function and with higher order for its derivatives. An infinitely di↵erentiable kernel function, smooth even for high order derivatives, such as the Gaussian, is a suitable choice to successfully provide any order of accuracy for the function approximation or its derivatives. However, the improved formulation requires many summations on the kernel function and its derivatives, strongly limiting the feasibility of the method. Motivated to speed up the computation and to make large scale problems tractable we adopt fast summation approach. Namely, the improved fast Gaussian transform has been opportunely adapted to assembly the corrective linear system for each evaluation point picking up the computational cost at an acceptable level preserving the accuracy. Numerical experiments with various bivariate functions and sets of data are reported to give evidence of the accuracy, convergency and computational demanding.

In this paper we propose for the first time an iterative approach of the Smoothed Particle Hydrodynamics (SPH) method. The method is widespread in many areas of science and engineering and despite its extensive application it su↵ers from several drawbacks due to inaccurate approximation at boundaries and at irregular interior regions. The presented iterative process improves the accuracy of the standard method by updating the initial estimates iterating on the residuals. It is appealing preserving the matrix-free nature of the method and avoiding to modify the kernel function. Moreover the process refines the SPH estimates and it is not a↵ected by disordered data distribution. We discuss on the numerical scheme and experiments with a bivariate test function and di↵erent sets of data validate the adopted approach.

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