Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. ACCURACY OF SPH <s> In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied. <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. ACCURACY OF SPH <s> Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks. <s> BIB002
SPH is usually considered a second-order precision method BIB001 . On the right side of (3) on the Taylor to start, only to retain the first derivative, you can obtain the following BIB002 : where r is the remainder of the Taylor expansion because the smooth kernel function satisfies the regularization and symmetry conditions. The above equation can be written as follows: However, this second-order accuracy may not be true for all situations. Liu et al.  analyzed a situation in which a particle is near the boundary of the computational region, the support domain of the particle intersects the computational region, and the support domain is truncated by the boundary. Neither regularization nor symmetry conditions can be satisfied. At this time, the SPH kernel approximation of the function no longer has second-order accuracy. In terms of kernel approximation, SPH has second-order accuracy. By using SPH to approximate the partial differential equation, the precision ultimately depends on the discrete form of the particle approximation, rather than the continuous form of kernel approximation  . At the same time, the distribution of particles, the choice of the smoothing function, and the calculation accuracy of smoothing length have considerable influence. In addition, the boundary processing method directly affects the SPH numerical simulation and calculation accuracy.