A new corrective scheme for SPH

A new corrective scheme for Smoothed Particle Hydrodynamics (SPH) is introduced which greatly improves its accuracy, particularly in regions of particle deficiency or when particles are irregularly distributed. The scheme is based on the Taylor expansion of the SPH kernel estimates. The corrective equations are derived up to the second derivative in an arbitrary number of dimensions. Test applications for the new scheme include the interpolation of functions and the numerical solution of the heat equation in one and two dimensions.