A Review and Mathematical Analysis of the Moving Particle Semi-Implicit Method for Incompressible Flows

Abstract

Computational fluid dynamics (CFD) methods based on mesh discretization, such as the finite element method (FEM) and finite volume method (FVM), have well-established mathematical foundations but face challenges in simulating flows with strongly deforming free surfaces, droplet breakup, and multiphase interactions due to mesh distortion. The Moving Particle Semi-Implicit (MPS) method is a meshfree approach based on a Lagrangian description, enabling the natural treatment of large deformations and complex free surfaces. Although MPS has been widely applied to the simulation of breaking waves and fluid-structure interactions, the mathematical foundations of its discrete operators still require systematic clarification. This paper analyzes the mathematical foundations of MPS, including discrete gradient and Laplacian operators, the pressure Poisson equation, consistency based on Taylor expansion, spectral properties, and energy dissipation of the discrete system. The results clarify the mathematical properties of MPS under assumptions concerning particle distribution and operator structure.