A geometric PIC discretization of Lie-Poisson brackets

Non-dissipative models of fluids are known to possess a Lie-Poisson Hamiltonian structure. In discretizing such brackets, one encounters a closure problem: given a finite representation of the fields, it is usually not the case that the dynamic evolution of those fields is prescribed only in terms of that finite dataset. Particle based representations (e.g. point-vortex methods) for the 2D Euler equations circumvent this difficulty, but suffer from slow convergence and an unwieldy distributional representation of the field. We present the "dual PIC" method for the 2D Euler equations, a method based on PIC methods from plasma physics but with a dual Galerkin representation in addition to the particle based representation. The two representations are related to each other through an L2 projection. Moreover, the error in this projection is conserved as a Casimir invariant of the flow. While this method is presented in the context of the 2D Euler equations, it holds promise as a general method of Lie-Poisson systems