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Discrete Lie Advection of Differential Forms

Mullen, P. and McKenzie, A. and Pavlov, D. and Durant, L. and Tong, Y. and Kanso, E. and Marsden, J. E. and Desbrun, M. (2011) Discrete Lie Advection of Differential Forms. Foundations of Computational Mathematics, 11 (2). pp. 131-149. ISSN 1615-3375. (In Press)

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In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite-volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan’s homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.

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Additional Information:© 2010 SFoCM. Published online: 08 September 2010. Communicated by Douglas Arnold and Peter Olver. This research was partially supported by NSF grants CCF-0811313/ 0811373/- 0936830/1011944, CMMI-0757106/0757123/¯0757092, IIS-0953096, and DMS-0453145, and by the Center for the Mathematics of Information at Caltech.
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Caltech Center for the Mathematics of InformationUNSPECIFIED
Subject Keywords:Discrete contraction; Discrete Lie derivative; Discrete differential forms; Finite-volume methods; Hyperbolic PDEs
Classification Code:Mathematics Subject Classification (2000): 35Q35; 51P05; 65M08
Record Number:CaltechAUTHORS:20100930-091445735
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20234
Deposited By: Tony Diaz
Deposited On:01 Oct 2010 22:31
Last Modified:06 Jul 2011 21:40

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