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Divergence-free vector fields and curl-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations, Maxwell's equations, the equations for magnetohydrodynamics, and surface reconstruction. In practice, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free or discretely curl-free. This field can then be interpolated to non-grid locations, which is required for many algorithms such as particle tracing or semi-Lagrangian advection. This interpolated field will not generally be divergence-free or curl-free in the analytic sense. In this work, we assume these fields are stored on a MAC grid layout and that the divergence and curl operators are discretized using finite differences. This work builds on and extends [39] in multiple ways: (1) we design a divergence-free interpolation scheme that preserves the discrete flux, (2) we adapt the general construction of divergence-free fields into a general construction for curl-free fields, (3) we extend the framework to a more general class of finite difference discretizations, and (4) we use this flexibility to construct fourth-order accurate interpolation schemes for the divergence-free case and the curl-free case. All of the constructions and specific schemes are explicit piecewise polynomials over a local neighborhood. 

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.