Advertisement

Foundations of Computational Mathematics

, Volume 8, Issue 4, pp 427–467 | Cite as

Difference Forms

  • Elizabeth L. MansfieldEmail author
  • Peter E. Hydon
Article

Abstract

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes.

Keywords

Difference forms Lattice variety Cohomology Difference chains Local exactness Local difference potentials 

Mathematics Subject Classification (2000)

39A12 14F40 55N05 15A75 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Arnold, Differential complexes and numerical stability, Plenary address at ICM Beijing, 2002. http://www.ima.umn.edu/~arnold/papers/icm2002.pdf.
  2. 2.
    D. Arnold, R. S. Falk, and R. Winthur, Finite element exterior calculus, homological techniques and applications, Acta Numerica 15 (2006), 1–155. zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Bossavit, Differential forms and the computation of fields and forces in electromagnetism, Eur. J. Mech. B-Fluids 10 (1991), 474–488. zbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Bossavit, Whitney forms—a class of finite-elements for 3-dimensional computations in electromagnetism, IEE Proc. A 135 (1988), 493–500. Google Scholar
  5. 5.
    R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York, 1982. zbMATHGoogle Scholar
  6. 6.
    T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that preserve symplecticity, Phys. Lett. A 284 (2001), 184–193. zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    C. J. Budd and G. J. Collins, An invariant moving mesh scheme for the nonlinear diffusion equation, App. Num. Math. 26 (1998), 23–29. zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C. J. Budd and M. D. Piggott, Geometric integration and its applications, in Handbook of Numerical Analysis, Vol. XI, pp. 35–139, North-Holland, Amsterdam, 2003. Google Scholar
  9. 9.
    J. A. Chard and V. Shapiro, A multivector data structure for differential forms and equations, Math. Comp. Sim. 54 (2000), 33–64. CrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Forman, Morse theory for cell complexes, Topology 37 (1998), 945–979. zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    T. Frankel, The Geometry of Physics, Cambridge University Press, Cambridge, 1997. zbMATHGoogle Scholar
  12. 12.
    W. Fulton, Algebraic Topology, A First Course, Graduate Texts in Mathematics, Vol. 153, Springer-Verlag, New York, 1995. zbMATHGoogle Scholar
  13. 13.
    G. Zhong and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133 (1988), 134–139. CrossRefMathSciNetGoogle Scholar
  14. 14.
    P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Publications, Vol. 48, Cambridge University Press, Cambridge, 2004. zbMATHGoogle Scholar
  15. 15.
    G. Chaohao (ed.), Soliton Theory and its Applications, Springer-Verlag, Berlin, 1995. zbMATHGoogle Scholar
  16. 16.
    E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, Springer-Verlag, Berlin, 2002. zbMATHGoogle Scholar
  17. 17.
    R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica 11 (2002), 237–340. zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    P. E. Hydon, Symmetries and first integrals of ordinary difference equations, Proc. Roy. Soc. A 456 (2000), 2835–2855. zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    P. E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A: Math. Gen. 34 (2001), 10347–10355. zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, J. Found. Comp. Math. 4 (2004), 187–217. zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Iserles, H. Munthe-Kaas, S. Norsett, and A. Zanna, Lie group methods, Acta Numerica 9 (2000), 215–365. CrossRefMathSciNetGoogle Scholar
  22. 22.
    T. Kaczynski, K. Mischaikow, and M. Mrozek, Computing homology, Homology, Homotopy and Applications 5 (2003), 233–256. zbMATHMathSciNetGoogle Scholar
  23. 23.
    L. Kharevych, W. Y. Tong, E. Kanso, J. E. Marsden, P. Schröder, and M. Desbrun Geometric, variational integrators for computer animation, in Eurographics/ACM SIGGRAPH Symposium on Computer Animation (M.-P. Cani and J. O’Brien, eds.), pp. 43–51, ACM, Portland, 2006. Google Scholar
  24. 24.
    T. Y. Kong, R. D. Kopperman, and P. R. Meyer, A topological approach to digital topology, Amer. Math. Monthly 98 (1991), 901–917. zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    M. Leok, Foundations of computational geometric mechanics, PhD Thesis, California Institute of Technology, 2004. Google Scholar
  26. 26.
    R. I. McLachlan and G. R. W. Quispel, Six lectures on the geometric integration of ODEs, in Foundations of Computational Mathematics (R. A. DeVore, A. Iserles, and E. Süli, eds.), London Mathematical Society Lecture Note Series, Vol. 284, pp. 155–210, Cambridge University Press, Cambridge, 2001. Google Scholar
  27. 27.
    E. L. Mansfield and P. E. Hydon, Towards approximations which preserve integrals, in Proc. ISSAC 2001 (B. Mourrain, ed.), pp. 217–222, ACM Publishing, New York, 2001. CrossRefGoogle Scholar
  28. 28.
    W. S. Massey, Singular Homology Theory, Graduate Texts in Mathematics, Vol. 70, Springer-Verlag, Berlin, 1980. zbMATHGoogle Scholar
  29. 29.
    C. Mattiussi, An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comp. Phys. 133 (1997), 289–309. zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    A. R. Mohebalhojeh, On shallow water potential vorticity inversion by Rossby-number expansions, Q. J. Roy. Met. Soc. 128 (2002), 679–694. CrossRefGoogle Scholar
  31. 31.
    J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994. zbMATHGoogle Scholar
  32. 32.
    W. Schwalm, B. Moritz, M. Giona, and M. Schwalm, Vector difference calculus for physical lattice models, Phys. Rev. E 59 (1999), 1217–1233. CrossRefMathSciNetGoogle Scholar
  33. 33.
    S. Suuriniemi and L. Kettunen, Trade-off between information and complexity: a technique for automated topological computations, COMPEL 22 (2003), 481–494. zbMATHMathSciNetGoogle Scholar
  34. 34.
    Y. Y. Tong, S. Lombeya, A. N. Hirani, and M. Desbrun, Discrete multiscale vector field decomposition, ACM Trans. Graphics 22 (2003), 445–452. CrossRefGoogle Scholar
  35. 35.
    E. Tonti, On the formal structure of physical theories, Istituto di Matematica del Politecnico di Milano, Milan, http://www.dic.univ.trieste.it/perspage/tonti/papershtm, 1975.
  36. 36.
    A. Weil, Sur les théorèmes de De Rham, Comment. Math. Hel. 26 (1952), 119–145. zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, 1957. zbMATHGoogle Scholar
  38. 38.
    D. K. Wise, p-form electromagnetism on discrete space-times, Class. Quantum Grav. 23 (2006), 5129–5176. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© SFoCM 2007

Authors and Affiliations

  1. 1.Institute of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  2. 2.Department of Mathematics and StatisticsUniversity of SurreyGuildfordUK

Personalised recommendations