New corrective scheme for DF-SPH

A new corrective scheme for the derivative-free (DF) smoothed particle hydrodynamics (SPH) method is proposed to compute the first- and second-order derivatives. To demonstrate the ability of the new corrective scheme, the first- and second-order derivatives of the given function were computed when the particles were uniformly and irregularly distributed. The nonlinear convection-diffusion problems were simulated using both the DF-SPH method and the proposed method in one- and two-dimensional cases, and root mean square errors were compared. Results show that numerical solutions acquired by the proposed corrective scheme agree very well with analytical solutions and that the new scheme is more efficient, accurate, and stable than the original DF-SPH.