Lie-Poisson integrators in Hamiltonian fluid mechanics

Citation

Ryan, Barry James (1993) Lie-Poisson integrators in Hamiltonian fluid mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/n4ds-zz40. https://resolver.caltech.edu/CaltechETD:etd-10242005-152235

Abstract

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This thesis explores the application of geometric mechanics to problems in 2D, incompressible, inviscid fluid mechanics. The main motivation is to try to develop symplectic integration algorithms to model the Hamiltonian structure of inviscid fluid flow. The main manifestation of this Hamiltonian or conservative nature is the preservation of the infinite family of Casimirs parametrized by the body integrals of vorticity in the 2D case. The main difficulties encountered in trying to model the Hamiltonian structure of a fluid mechanical system are that the configuration space for the Hamiltonian flow is an infinite dimensional Frechet space and that the phase space is not symplectic but Lie-Poisson. Therefore, an appropriate finite mode truncation must be constructed under the constraint that it too remains Poisson and in some sense converges to the infinite dimensional parent manifold. With such a truncation in hand, there still remains the obstacle of non-symplectic structure. This geometry invalidates the application of traditional symplectic integrators and requires a more sophisticated algorithm.

We develop a Lie-Poisson truncation on the Lie group SU(N) for the Euler equations on the special geometry of a twice periodic domain in [...]. We show that this finite dimensional analog is compatible with the Arnold[5] formulation of Hamiltonian mechanics on Lie groups with a left or right invariant metric. We then proceed to review the Lie-Poisson integration literature and to develop Hamilton-Jacobi type symplectic algorithms for a broad class of Lie groups. For this same class of groups, we also succeed in constructing an explicit Lie-Poisson algorithm which radically improves computational speed over the current implicit schema. We test this new algorithm against a Hamilton-Jacobi implicit technique with favorable results.

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