A Level Set Method on Particle Flow Maps

Accurate interface tracking is central to numerical simulations involving moving boundaries, including multiphase flows, free‐surface dynamics, phase‐change problems, and other interface‐driven processes in computational mechanics. Although, level set methods provide a flexible implicit representation of evolving interfaces, their grid‐based advection and reinitialization procedures often introduce numerical diffusion, volume loss, and degradation of thin or highly curved structures. This paper introduces a particle flow map level set (PFM‐LS) method that addresses these limitations by transporting interface geometry through particle‐carried bidirectional flow maps. In contrast to particle level set methods that use particles mainly as corrective markers, PFM‐LS uses particles as primary carriers of level‐set values, gradients, and Hessians within a narrow band around the interface. The method combines long‐range flow‐map transport, high‐order particle‐grid reconstruction, adaptive particle control, and a hybrid particle‐grid redistancing scheme to preserve both signed‐distance consistency and sub‐grid geometric features. Numerical benchmarks in two and three dimensions, including topology‐change tests, chamfer and Hausdorff shape‐error metrics, and coupling to an incompressible free‐surface flow solver, show that PFM‐LS achieves strong volume preservation, high shape fidelity, and near fourth‐order convergence, outperforming a broad range of existing level set and reference‐map‐based methods.