  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.
