The MPS method is used to solve the Navier-Stokes equations in a Lagrangian framework. A fractional step method is applied which consists of splitting each time step in two steps of prediction and correction. The fluid is represented with particles, and the motion of each particle is calculated based on the interactions with the neighboring particles by means of a kernel function <ref>{{Cite journal|last=Nabian|first=Mohammad Amin|last2=Farhadi|first2=Leila|title=Multiphase Mesh-Free Particle Method for Simulating Granular Flows and Sediment Transport|url=https://ascelibrary.org/doi/abs/10.1061/(ASCE)HY.1943-7900.0001275|journal=Journal of Hydraulic Engineering|language=en|volume=143|issue=4|pages=04016102|doi=10.1061/(asce)hy.1943-7900.0001275|year=2017}}</ref><ref>{{Cite journalbook|last=Nabian|first=Mohammad Amin|last2=Farhadi|first2=Leila|date=2014-08-03|titlechapter=Numerical Simulation of Solitary Wave Using the Fully Lagrangian Method of Moving Particle Semi Implicit|urlpages=https://dx.V01DT30A006|doi.org/=10.1115/FEDSM2014-22237|pagestitle=V01DT30A006Volume 1D, Symposia: Transport Phenomena in Mixing; Turbulent Flows; Urban Fluid Mechanics; Fluid Dynamic Behavior of Complex Particles; Analysis of Elementary Processes in Dispersed Multiphase Flows; Multiphase Flow with Heat/Mass Transfer in Process Technology; Fluid Mechanics of Aircraft and Rocket Emissions and Their Environmental Impacts; High Performance CFD Computation; Performance of Multiphase Flow Systems; Wind Energy; Uncertainty Quantification in Flow Measurements and Simulations|doiisbn=10.1115/FEDSM2014978-222370-7918-4624-7}}</ref><ref>{{Cite journalbook|last=Nabian|first=Mohammad Amin|last2=Farhadi|first2=Leila|date=2014-11-14|titlechapter=Stable Moving Particle Semi Implicit Method for Modeling Waves Generated by Submarine Landslides|urlpages=https://dx.V007T09A019|doi.org/=10.1115/IMECE2014-40419|pagestitle=V007T09A019Volume 7: Fluids Engineering Systems and Technologies|doiisbn=10.1115/IMECE2014978-404190-7918-4954-5}}</ref>. The MPS method is similar to the SPH ([[smoothed-particle hydrodynamics]]) method (Gingold and Monaghan, 1977; Lucy, 1977) in that both methods provide approximations to the strong form of the [[partial differential equations]] (PDEs) on the basis of integral interpolants. However, the MPS method applies simplified [[differential operator]] models solely based on a local [[Weighted average|weighted averaging]] process without taking the [[gradient]] of a kernel function. In addition, the solution process of MPS method differs to that of the original SPH method as the solutions to the PDEs are obtained through a semi-implicit prediction-correction process rather than the fully explicit one in original SPH method.
