Calculating the smoothing error in SPH

•A comprehensive study of the smoothing error in SPH operators.•Gives errors as formal differential operators as well as Taylor expansions.•Excellent validation vs numerical tests.•Shows that the SPH smoothing error is smaller than believed for harmonic functions.•Allows improvements in all theoretical studies of SPH stability, consistency, etc. The smoothed particle hydrodynamics (SPH) numerical method is based on two approximations: a smoothing interpolation and a discrete (particle-based) approximation. The smoothing error has been identified in the early years of SPH as being proportional to the square of the smoothing length in the absence of boundaries and using a radial normalised kernel. Here we calculate the exact smoothing error as a function of the kernel standard deviation as a differential operator applied to the interpolated field. The key feature is that this error depends on the kernel Laplace transform. This technique is applied to the main SPH smoothed differential operators (gradient, divergence, pressure Laplacian and viscous forces). The present theory is tested against numerical data for harmonic and Gaussian functions with an excellent agreement.