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

Licentiate thesis, 2018

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.