A novel numerical technique designed for interface flow simulations using the Volume of Fluid (VOF) method on arbitrary unstructured meshes has been introduced. The method is called SimPLIC, which seamlessly integrates Piecewise Linear Interface Calculation (PLIC) and Simpson's rule. The main focus of the proposed method is to compute the volume of the primary phase that moves across a mesh face within a single time step. This is achieved by reconstructing the interface and assessing how the submerged face area evolves over time. Simpson's rule is employed to integrate the time evolution of this submerged face area, ensuring an accurate estimation of the volume of the transported primary phase. The method's robustness was validated by solving a spherical interface advection problem in a non-uniform three-dimensional flow across unstructured meshes with diverse cell types and dimensions. Key metrics such as volume conservation, shape retention, friction boundedness and solving efficiency were meticulously monitored and juxtaposed. Numerical outcomes underscored the precision and adequacy of the PLIC-VOF technique when complemented with Simpson's rule in advecting the interface. Furthermore, the SimPLIC method has been integrated into OpenFOAM v2312 as an unofficial extension and is now accessible to the community.