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We consider a general discretization strategy for Hamiltonian field theoriesgenerated by Lie-Poisson brackets which we call dual PIC (DPIC). This methodinvolves prescribing two different discrete representations of the dynamicalvariable which are constrained as a Casimir invariant of the flow to coincidewith one another via an L2 projection throughout the entire simulation. Thisallows one to leverage the relative advantages of each discrete representation.We begin by describing DPIC as applied to a general Lie-Poisson system and thenprovide illustrative examples: the discretization of the two-dimensionalvorticity equations and the Vlasov-Poisson equation.

Dual PIC: a structure preserving method for discretizing Lie-Poisson brackets