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Structure-preserving discretization of incompressible fluids

Pavlov, D. and Mullen, P. and Tong, Y. and Kanso, E. and Marsden, J. E. and Desbrun, M. (2011) Structure-preserving discretization of incompressible fluids. Physica D, 240 (6). pp. 443-458. ISSN 0167-2789. http://resolver.caltech.edu/CaltechAUTHORS:20110309-092043242

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Abstract

The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin’s circulation theorem is viewed as a consequence of Noether’s theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical–analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite-dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin’s circulation theorem holds.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.physd.2010.10.012 DOIUNSPECIFIED
http://www.sciencedirect.com/science/article/B6TVK-51J3646-1/2/fe9b9c020c7f27f6d7218b4b9b4b0e03PublisherUNSPECIFIED
Additional Information:© 2010 Elsevier B.V. Received 19 December 2009; revised 3 August 2010; accepted 28 October 2010. Communicated by J. Bronski. Available online 23 November 2010. We thank Daryl Holm and Yann Brénier for helpful early discussions and input, Evan Gawlik for generating the energy plots, and Keenan Crane for generating 2D tests. This research was partially supported by NSF grants CMMI-0757106, CCF-0811373 & 1011944, IIS-0953096, and DMS-0453145.
Funders:
Funding AgencyGrant Number
NSFCMMI-0757106
NSFCCF-0811373
NSFCCF-1011944
NSFIIS-0953096
NSFDMS-0453145
Subject Keywords:Euler equations; Lie group variational integrators
Record Number:CaltechAUTHORS:20110309-092043242
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20110309-092043242
Official Citation:D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J.E. Marsden, M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena, Volume 240, Issue 6, 1 March 2011, Pages 443-458, ISSN 0167-2789, DOI: 10.1016/j.physd.2010.10.012. (http://www.sciencedirect.com/science/article/B6TVK-51J3646-1/2/fe9b9c020c7f27f6d7218b4b9b4b0e03)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:22723
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:09 Mar 2011 18:00
Last Modified:09 Mar 2011 18:00

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