Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 080, 20 pages      arXiv:1504.01953      https://doi.org/10.3842/SIGMA.2015.080
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

Decio Levi a, Luigi Martina b and Pavel Winternitz ac
a) Mathematics and Physics Department, Roma Tre University and Sezione INFN of Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
b) Dipartimento di Matematica e Fisica - Università del Salento and Sezione INFN of Lecce, Via per Arnesano, C.P. 193 I-73100 Lecce, Italy
c) Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada (permanent address)

Received April 01, 2015, in final form September 22, 2015; Published online October 02, 2015

Abstract
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.

Key words: Lie algebras of Lie groups; integrable systems; partial differential equations; discretization procedures for PDEs.

pdf (5124 kb)   tex (4498 kb)

References

  1. Adler V.E., Startsev S.Ya., On discrete analogues of the Liouville equation, Theoret. and Math. Phys. 121 (1999), 1484-1495, arXiv:solv-int/9902016.
  2. Bihlo A., Invariant meshless discretization schemes, J. Phys. A: Math. Theor. 46 (2013), 062001, 12 pages, arXiv:1210.2762.
  3. Bihlo A., Coiteux-Roy X., Winternitz P., The Korteweg-de Vries equation and its symmetry-preserving discretization, J. Phys. A: Math. Theor. 48 (2015), 055201, 25 pages, arXiv:1409.4340.
  4. Bihlo A., Nave J.-C., Invariant discretization schemes using evolution-projection techniques, SIGMA 9 (2013), 052, 23 pages, arXiv:1209.5028.
  5. Bourlioux A., Cyr-Gagnon C., Winternitz P., Difference schemes with point symmetries and their numerical tests, J. Phys. A: Math. Gen. 39 (2006), 6877-6896, math-ph/0602057.
  6. Budd C., Dorodnitsyn V., Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 34 (2001), 10387-10400.
  7. Dorodnitsyn V., Transformation groups in difference spaces, J. Sov. Math. 55 (1991), 1490-1517.
  8. Dorodnitsyn V., Applications of Lie groups to difference equations, Differential and Integral Equations and Their Applications, Vol. 8, CRC Press, Boca Raton, FL, 2011.
  9. Dorodnitsyn V., Kaptsov E., Kozlov R., Winternitz P., The adjoint equation method for constructing first integrals of difference equations, J. Phys. A: Math. Theor. 48 (2015), 055202, 32 pages, arXiv:1311.1597.
  10. Dorodnitsyn V., Kozlov R., A heat transfer with a source: the complete set of invariant difference schemes, J. Nonlinear Math. Phys. 10 (2003), 16-50, math.AP/0309139.
  11. Dorodnitsyn V., Kozlov R., Winternitz P., Lie group classification of second-order ordinary difference equations, J. Math. Phys. 41 (2000), 480-504.
  12. Dorodnitsyn V., Kozlov R., Winternitz P., Continuous symmetries of Lagrangians and exact solutions of discrete equations, J. Math. Phys. 45 (2004), 336-359, nlin.SI/0307042.
  13. Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, and fields, Graduate Texts in Mathematics, Vol. 93, 2nd ed., Springer-Verlag, New York, 1992.
  14. Floreanini R., Negro J., Nieto L.M., Vinet L., Symmetries of the heat equation on the lattice, Lett. Math. Phys. 36 (1996), 351-355.
  15. Floreanini R., Vinet L., Lie symmetries of finite-difference equations, J. Math. Phys. 36 (1995), 7024-7042.
  16. Floreanini R., Vinet L., Quantum symmetries of $q$-difference equations, J. Math. Phys. 36 (1995), 3134-3156.
  17. Grammaticos B., Ramani A., Painlevé equations, continuous, discrete and ultradiscrete, in Symmetries and Integrability of Difference Equations (Université de Montréal, Montréal, QC, June 8-21, 2008), London Mathematical Society Lecture Note Series, Vol. 381, Editors D. Levi, P.J. Olver, Z. Thomova, P. Winternitz, Cambridge University Press, Cambridge, 2011, 50-82.
  18. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  19. Grant T.J., Bespoke finite difference schemes that preserve multiple conservation laws, LMS J. Comput. Math. 18 (2015), 372-403.
  20. Hairer E., Lubich C., Wanner G., Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, Vol. 31, 2nd ed., Springer-Verlag, Berlin, 2006.
  21. Hydon P.E., Difference equations by differential equation methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2014.
  22. Iserles A., A first course in the numerical analysis of differential equations, 2nd ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.
  23. Levi D., Martina L., Winternitz P., Lie-point symmetries of the discrete Liouville equation, J. Phys. A: Math. Theor. 48 (2015), 025204, 18 pages, arXiv:1407.4043.
  24. Levi D., Olver P.J., Thomova Z., Winternitz P. (Editors), Symmetries and integrability of difference equations, London Mathematical Society Lecture Note Series, Vol. 381, Cambridge University Press, Cambridge, 2011.
  25. Levi D., Rodríguez M.A., Construction of partial difference schemes: I. The Clairaut, Schwarz, Young theorem on the lattice, J. Phys. A: Math. Theor. 46 (2013), 295203, 19 pages.
  26. Levi D., Rodríguez M.A., On the construction of partial difference schemes: II. Discrete variables and invariant schemes, arXiv:1407.0838.
  27. Levi D., Vinet L., Winternitz P., Lie group formalism for difference equations, J. Phys. A: Math. Gen. 30 (1997), 633-649.
  28. Levi D., Winternitz P., Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991), 335-338.
  29. Levi D., Winternitz P., Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006), R1-R63, nlin.SI/0502004.
  30. Levi D., Winternitz P., Yamilov R.I., Lie point symmetries of differential-difference equations, J. Phys. A: Math. Theor. 43 (2010), 292002, 14 pages, arXiv:1004.5311.
  31. Lie S., Theorie der Transformationsgruppen, Bds. 1-3, B.G. Teubner, Leipzig, 1888, 1890, 1893.
  32. Liouville J., Sur l'equation aux différences partielles $\frac{d^2 \log \lambda}{d ud v} \pm \frac{\lambda}{2a^2}=0$, J. Math. Pures Appl. 18 (1853), 71-72.
  33. Marsden J.E., West M., Discrete mechanics and variational integrators, Acta Numer. 10 (2001), 357-514.
  34. McLachlan R.I., Quispel G.R.W., Geometric integrators for ODEs, J. Phys. A: Math. Gen. 39 (2006), 5251-5285.
  35. Medolaghi P., Classificazione delle equazioni alle derivate parziali del secondo ordine, che ammettono un gruppo infinito di trasformazioni puntuali, Ann. Mat. Pura Appl. 1 (1898), 229-263.
  36. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  37. Rebelo R., Valiquette F., Symmetry preserving numerical schemes for partial differential equations and their numerical tests, J. Difference Equ. Appl. 19 (2013), 738-757, arXiv:1110.5921.
  38. Rebelo R., Valiquette F., Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups, J. Difference Equ. Appl. 21 (2015), 285-318, arXiv:1401.4380.
  39. Rebelo R., Winternitz P., Invariant difference schemes and their application to ${\rm sl}(2,{\mathbb R})$ invariant ordinary differential equations, J. Phys. A: Math. Theor. 42 (2009), 454016, 10 pages, arXiv:0906.2980.
  40. Rodríguez M.A., Winternitz P., Lie symmetries and exact solutions of first-order difference schemes, J. Phys. A: Math. Gen. 37 (2004), 6129-6142, nlin.SI/0402047.
  41. Valiquette F., Winternitz P., Discretization of partial differential equations preserving their physical symmetries, J. Phys. A: Math. Gen. 38 (2005), 9765-9783, math-ph/0507061.
  42. Winternitz P., Symmetry preserving discretization of differential equations and Lie point symmetries of differential-difference equations, in Symmetries and Integrability of Difference Equations (Université de Montréal, Montréal, QC, June 8-21, 2008), London Mathematical Society Lecture Note Series, Vol. 381, Editors D. Levi, P.J. Olver, Z. Thomova, P. Winternitz, Cambridge University Press, Cambridge, 2011, 292-341.


Previous article  Next article   Contents of Volume 11 (2015)