Symplectic methods for Hamiltonian isospectral flows and 2D incompressible Euler equations on a sphere

Sammanfattning: 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. In this thesis, I develop a new class of numerical schemes for Hamiltonian isospectral flows, in order to solve the hydrodynamical Euler equations on a sphere. The results are presented in two papers. In the first one, we derive a general framework for the isospectral flows, providing then a class of numerical methods of arbitrary order, based on the Lie–Poisson reduction of Hamiltonian systems. Avoiding the use of any constraint, we obtain a large class of numerical schemes for Hamil- tonian and non-Hamiltonian isospectral flows. One of the advantages of these methods is that, together with the isospectrality, they have near conservation of the Hamiltonian and, indeed, they are Lie–Poisson inte- grators. In the second paper, using the results of the first one, 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.