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Structure-preserving discretization of incompressible fluids


Pavlov, Dmitry (2009) Structure-preserving discretization of incompressible fluids. Dissertation (Ph.D.), California Institute of Technology.


The geometric nature of Euler fluids has been clearly identified and extensively studied in mathematics. However computational approaches to fluid mechanics, mostly derived from a numerical-analytic point of view, are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts. In contrast, we geometrically derive discrete equations of motion for fluid dynamics from first principles. Our approach uses a finite-dimensional Lie group to discretize the group of volume-preserving diffeomorphisms, and the discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion induce a structure-preserving time integrator with good long-term energy behavior, for which an exact discrete Kelvin circulation theorem holds. Possible extensions of our method to magnetohydrodynamics, viscous flows, optimal transport and a link to Brenier's generalized flows are also discussed.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Discrete Exterior Calculus; Euler equations; Ideal Fluid; Lin Constraints; Non-holonomic constraints; Variational Integrators
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marsden, Jerrold E. (advisor)
  • Desbrun, Mathieu (advisor)
Thesis Committee:
  • Marsden, Jerrold E. (chair)
  • Breuer, Jonathan
  • Desbrun, Mathieu
  • Makarov, Nikolai G.
Defense Date:13 May 2009
Record Number:CaltechETD:etd-05222009-125630
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1963
Deposited By: Imported from ETD-db
Deposited On:29 May 2009
Last Modified:26 Dec 2012 02:44

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