A method to improve first order approximation of smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) is a useful meshless method. The first and second orders are the most popular derivatives of the field function in the mechanical governing equations. New methods were proposed to improve accuracy of SPH approximation by the lemma proved. The lemma describes the relationship of functions and their SPH approximation. Finally, the error comparison of SPH method with or without our improvement was carried out.