Finite dimensional lie‐poisson approximations to vlasov‐poisson equations

Source abstract

Abstract Many of the basic equations of conservative continuum mechanics (Euler, Vlasov‐Poisson, Vlasov‐Maxwell, MHD, etc.) are Hamiltonian systems with respect to Lie‐Poisson brackets on dual spaces of infinite dimensional Lie algebras. The development of Lie‐Poisson integrators for finite dimensional Lie‐Poisson systems has shown that they are superior in the numerical simulations of these systems, especially with regard to long term phenomena. This paper shows how to truncate one of these systems, the Vlasov‐Poisson equation of plasma physics, to a finite dimensional Lie‐Poisson system. This requires replacing the functions on single‐particle phase space, with their Poisson bracket Lie algebra structure, by a finite dimensional Lie algebra. Replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their Taylor expansions up to order k. Unfortunately, these truncated Taylor expansions do not form a Lie algebra, since the functions which vanish through order k do not form an ideal under Poisson bracket. Geometrically, this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup. Introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the “location” from “shape” coordinates of a lump of density in phase space, as far as the Poisson bracket is concerned. One can then replace the shape coordinates by a finite number of moments. The central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a Lie group as a product of subgroups. The main theorem is first proved by the general theory of Poisson reduction, then by explicit calculation, and lastly by showing that the Poisson isomorphism follows from the lift of a natural groupoid isomorphism. The groupoid aspect of the theory also provides natural Poisson maps, useful in the application of Ruth‐type integration techniques, which do not seem easily derivable from the general theory of Poisson reduction. © 1994 John Wiley & Sons, Inc.