Foundations of Computational Mathematics

, Volume 11, Issue 2, pp 131–149 | Cite as

Discrete Lie Advection of Differential Forms

  • P. Mullen
  • A. McKenzie
  • D. Pavlov
  • L. Durant
  • Y. Tong
  • E. Kanso
  • J. E. Marsden
  • M. Desbrun
Article

Abstract

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.

Keywords

Discrete contraction Discrete Lie derivative Discrete differential forms Finite-volume methods Hyperbolic PDEs 

Mathematics Subject Classification (2000)

35Q35 51P05 65M08 

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Copyright information

© SFoCM 2010

Authors and Affiliations

  • P. Mullen
    • 1
  • A. McKenzie
    • 1
  • D. Pavlov
    • 1
  • L. Durant
    • 1
  • Y. Tong
    • 2
  • E. Kanso
    • 3
  • J. E. Marsden
    • 1
  • M. Desbrun
    • 1
  1. 1.Computing + Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Computer Science & EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Aerospace and Mechanical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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