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Symplectic methods for Hamiltonian isospectral flows and 2D incompressible Euler equations on a sphere
Licentiate thesis, 2018

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.

Fluid dynamics.

Symplectic methods

Isospectral flows

Geometric integration

Hamiltonian systems

Structure preserving algorithms

Euler equations

Lie–Possion systems

## Author

### Milo Viviani

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

### Subject Categories

Mathematics

### Publisher

Chalmers University of Technology

Euler, Matematiska vetenskaper, Chalmers tvärgata 3, Göteborg

Opponent: Antonella Zanna, Department of Mathematics, University of Bergen, Norway.