Skip to main content
SpringerLink
Log in

Discrete Lie Advection of Differential Forms

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

An Erratum to this article was published on 12 February 2011

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75 (Springer, Berlin, 1988).

    MATH  Google Scholar 

  2. D.N. Arnold, P.B. Bochev, R.B. Lehoucq, R.A. Nicolaides, M. Shashkov (eds.), Compatible Spatial Discretizations. IMA Volumes, vol. 142 (Springer, Berlin, 2006).

    MATH  Google Scholar 

  3. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc. 47, 281–354 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. P.B. Bochev, J.M. Hyman, Principles of mimetic discretizations of differential operators, IMA 142, 89–119 (2006).

    MathSciNet  Google Scholar 

  6. A. Bossavit, Computational Electromagnetism (Academic Press, Boston, 1998).

    MATH  Google Scholar 

  7. A. Bossavit, Extrusion contraction: their discretization via Whitney forms, COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 22(3), 470–480 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  8. W.L. Burke, Applied Differential Geometry (University Press, Cambridge, 1985).

    MATH  Google Scholar 

  9. S. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Pearson Education, Upper Saddle River, 2003).

    Google Scholar 

  10. É. Cartan, Les Systèmes Differentiels Exterieurs et leurs Applications Géometriques (Hermann, Paris, 1945).

    MATH  Google Scholar 

  11. M. Desbrun, E. Kanso, Y. Tong, Discrete differential forms for computational sciences, in Discrete Differential Geometry, Course Notes, ed. by E. Grinspun, P. Schröder, M. Desbrun (ACM SIGGRAPH, New York, 2006).

    Google Scholar 

  12. T.F. Dupont, Y. Liu, Back-and-forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function, J. Comput. Phys. 190(1), 311–324 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Dyadechko, M. Shashkov, Moment-of-Fluid Interface Reconstruction. LANL Technical Report LA-UR-05-7571 (2006).

  14. S. Elcott, Y. Tong, E. Kanso, P. Schröder, M. Desbrun, Stable circulation-preserving, simplicial fluids, ACM Trans. Graph. 26(1), 4 (2007).

    Article  Google Scholar 

  15. B. Engquist, S. Osher, One-sided difference schemes and transonic flow, PNAS 77(6), 3071–3074 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Flanders, Differential Forms and Applications to Physical Sciences (Dover, New York, 1990).

    Google Scholar 

  17. T. Frankel, The Geometry of Physics, 2nd edn. (Cambridge University Press, Cambridge, 2004).

    MATH  Google Scholar 

  18. X. Gu, S.T. Yau, Global conformal surface parameterization, in Symposium on Geometry Processing (2003), pp. 127–137.

  19. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for ODEs (Springer, Berlin, 2002).

    Google Scholar 

  20. F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surfaces, Phys. Fluids 8, 2182–2189 (1965).

    Article  MATH  Google Scholar 

  21. H. Heumann, R. Hiptmair, Extrusion contraction upwind schemes for convection-diffusion problems, in Seminar für Angewandte Mathematik SAM 2008-30, ETH Zürich (2008).

  22. D.J. Hill, D.I. Pullin, Hybrid tuned center-difference-WENO method for large Eddy simulations in the presence of strong shocks, J. Comput. Phys. 194(2), 435–450 (2004).

    Article  MATH  Google Scholar 

  23. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11, 237–339 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  24. A.N. Hirani, Discrete exterior calculus. Ph.D. thesis, Caltech (2003).

  25. A. Iske, M. Käser, Conservative semi-Lagrangian advection on adaptive unstructured meshes, Numer. Methods Partial Differ. Equ. 20(3), 388–411 (2004).

    Article  MATH  Google Scholar 

  26. R.J. LeVeque, CLAWPACK, at http://www.clawpack.org (1994–2009).

  27. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002).

    Book  Google Scholar 

  28. D. Levy, S. Nayak, C. Shu, Y.T. Zhang, Central WENO schemes for Hamilton–Jacobi equations on triangular meshes, J. Sci. Comput. 27, 532–552 (2005).

    MATH  Google Scholar 

  29. X.D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Sci. Comput. 126, 202–212 (1996).

    Google Scholar 

  30. D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Dover, New York, 1993).

    Google Scholar 

  31. J.E. Marsden, M. West, Discrete mechanics and variational integrators, Acta Numer. (2001).

  32. S. Morita, Geometry of Differential Forms. Translations of Mathematical Monographs, vol. 201 (Am. Math. Soc., Providence, 2001).

    MATH  Google Scholar 

  33. J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Menlo Park, 1984).

    MATH  Google Scholar 

  34. J.C. Nédélec, Mixed Finite Elements in 3D in H(div) and H(curl). Springer Lectures Notes in Mathematics, vol. 1192 (Springer, Berlin, 1986).

    Google Scholar 

  35. R.A. Nicolaides, X. Wu, Covolume solutions of three dimensional div–curl equations, SIAM J. Numer. Anal. 34, 2195 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153 (Springer, New York, 2003).

    MATH  Google Scholar 

  37. J.A. Sethian, Level Set Methods and Fast Marching Methods, 2nd edn. Monographs on Appl. Comput. Math., vol. 3 (Cambridge University Press, Cambridge, 1999).

    MATH  Google Scholar 

  38. J. Shi, C. Hu, C.W. Shu, A technique for treating negative weights in WENO schemes, J. Comput. Phys. 175, 108–127 (2002).

    Article  MATH  Google Scholar 

  39. C.W. Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. Lecture Notes in Mathematics, vol. 1697 (Springer, Berlin, 1998), pp. 325–432.

    Google Scholar 

  40. C.W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Sci. Comput. 77, 439–471 (1988).

    MathSciNet  MATH  Google Scholar 

  41. A. Stern, Y. Tong, M. Desbrun, J.E. Marsden, Variational integrators for Maxwell’s equations with sources, in Progress in Electromagnetics Research Symposium, vol. 4 (2008), pp. 711–715.

  42. V.A. Titarev, E.F. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys. 201(1), 238–260 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  43. Y. Tong, P. Alliez, D. Cohen-Steiner, M. Desbrun, Designing quadrangulations with discrete harmonic forms, in Proc. Symp. Geometry Processing (2006), pp. 201–210.

  44. H. Whitney, Geometric Integration Theory (Princeton Press, Princeton, 1957).

    MATH  Google Scholar 

  45. Y.T. Zhang, C.W. Shu, High-order WENO schemes for Hamilton–Jacobi equations on triangular meshes, J. Sci. Comput. 24, 1005–1030 (2003).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA

    P. Mullen, A. McKenzie, D. Pavlov, L. Durant, J. E. Marsden & M. Desbrun

  2. Computer Science & Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Y. Tong

  3. Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    E. Kanso

Authors
  1. P. Mullen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. McKenzie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Pavlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Durant
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Y. Tong
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. E. Kanso
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. J. E. Marsden
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. M. Desbrun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Desbrun.

Additional information

Communicated by Douglas Arnold and Peter Olver.

An erratum to this article can be found at http://dx.doi.org/10.1007/s10208-011-9089-1

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mullen, P., McKenzie, A., Pavlov, D. et al. Discrete Lie Advection of Differential Forms. Found Comput Math 11, 131–149 (2011). https://doi.org/10.1007/s10208-010-9076-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-010-9076-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation