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Geometric, variational discretization of continuum theories

Gawlik, E. S. and Mullen, P. and Pavlov, D. and Marsden, J. E. and Desbrun, M. (2011) Geometric, variational discretization of continuum theories. Physica D, 240 (21). pp. 1724-1760. ISSN 0167-2789.

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This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler–Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.

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Additional Information:© 2011 Elsevier B.V. Received 24 October 2010; revised 25 June 2011; Accepted 19 July 2011. Communicated by B. Sandstede. Available online 28 July 2011. During the summer of 2009, Evan Gawlik was supported by the Caltech Summer Undergraduate Research Fellowship Program and the Aerospace Corporation.We wish to thank Francois Gay-Balmaz for his many valuable comments on the geometric formulation of complex fluid flow, and for his help in editing this paper. Evan Gawlik also wishes to acknowledge the support of the DOE Computational Science Graduate Fellowship program, which is provided under grant number DE-FG02-97ER25308.
Funding AgencyGrant Number
Department of Energy (DOE) Computational Science Graduate Fellowship ProgramDE-FG02-97ER25308
Caltech Summer Undergraduate Research Fellowship ProgramUNSPECIFIED
Aerospace CorporationUNSPECIFIED
Subject Keywords:Fluid dynamics; Magnetohydrodynamics; Complex fluids; Geometric discretization; Structure-preserving schemes
Issue or Number:21
Record Number:CaltechAUTHORS:20120103-140434914
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Official Citation:E.S. Gawlik, P. Mullen, D. Pavlov, J.E. Marsden, M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, Volume 240, Issue 21, 15 October 2011, Pages 1724-1760, ISSN 0167-2789, 10.1016/j.physd.2011.07.011.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28626
Deposited By: Tony Diaz
Deposited On:04 Jan 2012 17:53
Last Modified:03 Oct 2019 03:34

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