An Impulse-formed Navier-Stokes Solver based on Long-range Particle Flow Maps

We present a particle-grid characteristic-mapping framework that extends long-range characteristic mapping from inviscid flows to general Navier-Stokes dynamics with viscosity, body forces, and complex boundaries. Unlike traditional grid-based and vorticity-centered characteristic methods, our method is built on the observation that particle trajectories naturally provide the long-range flow map, enabling geometric quantities and their gradients to be transported in a direct and effective manner. We identify the impulse, the gauge variable of the velocity field, as the primary quantity mapped along characteristics while remaining compatible with standard velocity-based incompressible solvers. Using the 1-form representation of the impulse equation, we derive an integral formulation that decomposes the impulse evolution into a component transported geometrically along the particle flow map and a complementary component generated by viscosity and body forces evaluated through path integrals accumulated along particle trajectories. These components together yield a unified characteristic-mapping solver capable of handling incompressible Navier-Stokes flows with viscosity and body forces while maintaining the accuracy and geometric fidelity of characteristic transport.

💡 Research Summary

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The paper introduces a novel particle‑grid hybrid solver for the incompressible Navier‑Stokes equations that leverages long‑range particle flow maps and the impulse (also called the gauge‑transformed velocity) as the primary transported quantity. Traditional characteristic‑mapping methods have been limited to inviscid, force‑free Euler flows because they rely on a mapping of geometric quantities that cannot directly accommodate viscosity or body forces, and because they store the map on Eulerian grids, which introduces interpolation errors especially under strong deformation.

The authors observe that each Lagrangian particle naturally carries its own mapping from the initial configuration to the current one simply by integrating its trajectory. By attaching the impulse ( \mathbf{m}= \mathbf{u} + \nabla \phi ) to each particle, the method obtains a variable that satisfies a clean characteristic evolution law (the Lie derivative form) and is not constrained by the incompressibility condition. Using a differential‑form (1‑form) representation of the impulse equation, the Navier‑Stokes system is rewritten as

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