Fluid Implicit Particles on Coadjoint Orbits

TL;DR

The paper introduces CO-FLIP, a high-order, structure-preserving fluid simulator that extends FLIP with divergence-free mimetic interpolation, exact P2G/G2P reciprocity, and a Lie-group time integrator to enforce energy and circulation conservation on coadjoint orbits. Grounding the method in geometric fluid mechanics, it uses a momentum-map framework to couple a continuous Euler flow with a finite-dimensional discretization, yielding an I-discrete Euler flow that preserves the coadjoint orbit and Casimirs. Mimetic B-spline interpolation provides pointwise divergence-free grid velocities and a weakly exact pressure projection, enabling high-order accuracy without global solves in the advection step. A two-level time integration combines an explicit advection with an implicit trapezoidal step, augmented by an energy-based correction to maintain invariants even with discretization error. Numerical experiments demonstrate remarkable long-time stability, energy and helicity/Casimirs conservation, and the ability to produce detailed turbulent structures at relatively coarse grid resolutions, albeit with higher computational cost compared to traditional methods. The work paves the way for scalable, high-fidelity, structure-preserving fluid simulations and suggests directions for extending CO-FLIP to free-surface flows and further solver optimizations.

Abstract

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.

Figures (28)

• Figure 1: Pyroclastic cloud simulated using our method (CO-FLIP). Note the intricate vortical structures present on the plume despite the low simulation grid resolution of $96\times192\times96$. Inset: Image of volcano eruption from Mount St. Helens, Washington, USA, on May 18th, 1980 (Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons).
• Figure 2: The classical FLIP method for simulating incompressible fluids. Particles (top row) are moved, and their momentum updated, using a background velocity field which is reconstructed from the pressure projection of the transferred particle data to the grid (bottom row). The CO-FLIP algorithm modifies each stage: The particles move and their velocity covector is transformed (i) using a point-wise div-free interpolation (ii) of grid velocity data transferred from particles using the pseudoinverse of the div-free interpolation (iii), and, further, Galerkin pressure projected (iv). This time evolution is now a Hamiltonian ODE which we integrate geometrically (v).
• Figure 3: The (1,5)-torus unknot. They are referred to as unknot as the numbers 1 and 5 are no longer coprime Maggioni:2010:VEH. With time, the unknot moves forward and stretches apart until the vortex reconnection event happens. Our method at a low resolution of $64\times64\times64$ captures this phenomenon.
• Figure 4: Comparison of torus (1,5)-unknot experiment. Note that our method conserves energy and vortical structures throughout the simulation, while traditional methods lose both energy and helicity.
• Figure 5: A nozzle shooting out smoke and hitting a Spot-shaped Crane:2013:RFC obstacle. Note the intricate vortical structures our method (CO-FLIP) can create at a low resolution of $128\times64\times64$.

Theorems & Definitions (93)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof