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Functions and conservation as well as subsidiary equations in Level Set Method (LSM) are presented. After the mathematical formulation, improvements in the numerical methodology for LSM are reviewed here for advection schemes, reinitialization methods, hybrid methods, adaptive-grid LSM, dual-resolution LSM, sharp-interface LSM, conservative LSM, parallel computing and extension from two to multi fluid/phase as well as to various types of two-phase flow. In the second part of this article, LSM method based Computational Multi-Fluid Dynamics (CMFD) applications and analysis are reviewed for four different types of multi-phase flow: separated and parallel internal flow, drop/bubble dynamics during jet break-up, drop impact dynamics on a solid or liquid surface and boiling. In the last twenty years, LSM has established itself as a method which is easy to program and is accurate as well as computationally-efficient.

Level set method for computational multi-fluid dynamics: A review on developments, applications and analysis