Symplectic methods for isospectral flows and 2D ideal hydrodynamics
Doktorsavhandling, 2020
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.
Symplectic methods
Fluid dynamics
Integrability theory
Euler equations
Lie--Possion systems
Isospectral flows
Geometric integration
Structure preserving algorithms
Hamiltonian systems
Författare
Milo Viviani
Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik
Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey
Arnold Mathematical Journal,;Vol. 7(2021)p. 357-385
Artikel i vetenskaplig tidskrift
A Casimir preserving scheme for long-Time simulation of spherical ideal hydrodynamics
Journal of Fluid Mechanics,;Vol. 884(2020)
Artikel i vetenskaplig tidskrift
Lie–Poisson Methods for Isospectral Flows
Foundations of Computational Mathematics,;Vol. 20(2020)p. 889-921
Artikel i vetenskaplig tidskrift
A minimal-variable symplectic method for isospectral flows
BIT Numerical Mathematics,;Vol. 60(2020)p. 741-758
Artikel i vetenskaplig tidskrift
Ämneskategorier
Algebra och logik
Beräkningsmatematik
Farkostteknik
Matematisk analys
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie
Utgivare
Chalmers
Euler room - Matematiska Vetenskaper - Chalmers // PLEASE, CONTACT ME FOR THE PASSWORD OF THE ZOOM MEETING
Opponent: Prof. Jason Frank, Utrecht University, Netherlands