My PhD thesis

Structure-preserving discretization of incompressible fluids

Advisors: Professor Jerrold E. Marsden, Professor Mathieu Desbrun

Abstract: 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.

Full text can be downloaded from the website of Caltech Library.
© 2012 Dmitry Pavlov, Imperial College London Contact Me