Dual PIC: A Geometric PIC discretization of Lie-Poisson brackets

Virtually all non-dissipative models in plasma physics, from the Liouville equations and the BBGKY hierarchy to various kinetic and fluid models, have been shown to possess a Lie-Poisson structure when modeled as noncanonical Hamiltonian systems. In discretizing such brackets, one encounters a closure problem. That is, 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 circumvent this difficulty with relative ease, but typically suffer from limited accuracy and difficulties in coupling to grid-based variables. We present the "dual PIC" method for the Vlasov-Poisson system. This method makes explicit use of two different representations of the phase space density: a particle based discretization and a Galerkin representation. The two representations are related to each other through an L2 projection. Moreover, the error in this L2 projection is conserved as a Casimir invariant of the flow. While we present the method in the context of Vlasov-Poisson, the strategy holds promise for application to general Lie-Poisson brackets.