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

2026-01-31

AI Summary

This work addresses the challenge of simultaneously achieving long-range transport accuracy and geometric fidelity in simulating incompressible Navier–Stokes flows with viscosity, body forces, and complex boundaries—a limitation inherent in traditional methods. The authors propose a novel particle-trajectory-based framework for long-range characteristic mapping, which, for the first time, transports the impulse 1-form as the primary variable along flow trajectories in a geometrically consistent manner. Viscous and body force effects are incorporated exactly via path integrals. By integrating a particle-in-cell hybrid strategy with compatibility techniques for incompressible flow solvers, the method seamlessly handles complex boundary conditions, enabling high-accuracy, geometrically faithful simulations of incompressible fluid motion.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
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Abstract

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.

Problem

• Navier-Stokes
• characteristic mapping
• impulse
• viscosity
• incompressible flow

Innovation

• impulse
• characteristic mapping
• Navier-Stokes solver
• particle flow map
• geometric transport