An algorithm to improve consistency in Smoothed Particle Hydrodynamics

An algorithm to improve the numerical evaluation of derivatives of a field function in Smoothed Particle
Hydrodynamics is proposed. The algorithm is based on the solution, at each particle location, of a linear
system whose unknowns are the first three derivatives of the desired function; the coefficients of the linear
system are obtained from various possible particle approximations of the Taylor series expansion of
the function. The method proves to be 2nd-order consistent for the 1st derivatives and 1st-order consistent
for the 2nd derivatives, both inside the domain and close to the boundaries, and it is not affected by
an irregular particle distribution. A numerical test performed on the SPH solution of the viscous Burgers
equation proves that the method can be validly applied to the simulation of convection–diffusion
problems.