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.