Difference between smoothed particle hydrodynamics and moving particle semi-implicit operators

Smoothed particle hydrodynamics (SPH) and moving semi-implicit (MPS) methods are representative meshfree used to compute Lagrangian mechanics. The approximations of differential operators in the SPH MPS have several similarities, but theoretical discussion difference between them is limited. This study mathematically describes via a comprehensive derivation first- second-order derivative for each method. indicates that consistent with pressure Poisson equation least-squares approximation, respectively. variation consistency can explain schemes incompressible flow problem. Additionally, result novel anisotropic operators. strengthens understanding facilitates selection appropriate method by users. Furthermore, proposed contribute developing adaptive or multiscale distributions.