Symplectic methods for isospectral flows and 2D ideal hydrodynamics

Abstract: The numerical solution of non-canonical Hamiltonian systems is an active and still growing field of research. At the present time, the biggest challenges concern the realization of structure preserving algorithms for differential equations on infinite dimensional manifolds. Several classical PDEs can indeed be set in this framework, and in particular the 2D hydrodynamical Euler equations. In this thesis, I have developed a new class of numerical schemes for Hamiltonian and non-Hamiltonian isospectral flows, in order to solve the 2D hydrodynamical Euler equations. The use of a conservative scheme has revealed new insights in the 2D ideal hydrodynamics, showing clear connections between geometric mechanics, statistical mechanics and integrability theory. The results are presented in four papers. In the first paper, we derive a general framework for the isospectral flows, providing a new class of numerical methods of arbitrary order, based on the Lie--Poisson reduction of Hamiltonian systems. Avoiding the use of any constraint, we obtain geometric integrators for a large class of Hamiltonian and non-Hamiltonian isospectral flows. One of the advantages of these methods is that, together with the isospectrality, they exhibit near conservation of the Hamiltonian and, indeed, they are Lie--Poisson integrators. In the second paper, using the results of paper I and III, we present a numerical method based on the geometric quantization of the Poisson algebra of the smooth functions on a sphere, which gives an approximate solution of the Euler equations with a number of discrete first integrals which is consistent with the level of discretization. The conservative properties of these schemes have allowed a more precise analysis of the statistical state of a fluid on a sphere. On the one hand, we show the link of the statistical state with some conserved quantities, on the other hand, we suggest a mechanism of formation of coherent structures related to the integrability theory of point-vortices. In the third paper, I present and analyse a minimal variable isospectral Lie--Poisson integrator for quadratic matrix Lie algebras. This result comes from a more careful analysis of the isospectral midpoint method derived in paper I. I also present a detailed description of quadratic Lie algebras, showing under which conditions the related Lie--Poisson systems are also isospectral flows. In the fourth paper, we give a survey on the integrability theory of the point-vortex dynamics. In particular, we show that all the results found in literature can be derived in the framework of symplectic reduction theory. Furthermore, our work aims to connect the 2D Euler equations with the point-vortex dynamics, as suggested in paper II.