American Institute of Mathematical Sciences

March  2014, 34(3): 1229-1249. doi: 10.3934/dcds.2014.34.1229

Generating functions and volume preserving mappings

 1 Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway, Norway

Received  November 2012 Revised  February 2013 Published  August 2013

In this paper, we study generating forms and generating functions for volume preserving mappings in $\mathbf{R}^n$. We derive some parametric classes of volume preserving numerical schemes for divergence free vector fields. In passing, by extension of the Poincaré generating function and a change of variables, we obtained symplectic equivalent of the theta-method for differential equations, which includes the implicit midpoint rule and symplectic Euler A and B methods as special cases.
Citation: Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229
References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer, GMT 60, second edition, 1989. [2] M. M. Carroll, A representation theorem for volume preserving transformations, J. International Journal of Non-Linear Mechanics, 39 (2004), 219-224. doi: 10.1016/S0020-7462(02)00167-1. [3] P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems, IMA Journal of Numerical Analysis, 27 (2007), 381-–405. doi: 10.1093/imanum/drl039. [4] K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math, 4 (1986), 279-289. [5] K. Feng and Z.-J Shang, Volume-preserving algorithms for source-free dynamical systems, Numerische Mathematik, 71 (1995), 451-463. doi: 10.1007/s002110050153. [6] K. Feng and D.-L Wang, Dynamical systems and geometric construction of algorithms, Contemporary Mathematics, AMS, Providence, 163 (1994), 1-32. doi: 10.1090/conm/163/01547. [7] K. Feng, H. M. Wu, M.-Z. Qin and D.-L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comp. Math., 7 (1989), 71-96. [8] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations," Springer, Second edition, 2006. [9] A. Iserles, G. R. W. Quispel and P. S. P. Tse, B-series methods cannot be volume-reserving, BIT Numerical Mathematics, 47 (2007), 351-378. doi: 10.1007/s10543-006-0114-8. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [11] H. E. Lomeli and J. D. Meiss, Generating forms for exact volume preserving maps, Discrete and Continuous Dynamical Systems serie S, 2 (2009), 361-377. doi: 10.3934/dcdss.2009.2.361. [12] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [13] J. Moser, "Notes on Dynamical System," Courant Lecture Notes in Mathematics, AMS, New York, 2005. [14] J. Moser and A. P. Veselov, Discrete version of some classical integrable systems and factorization of matrix polynomials, Commun. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [15] J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, 1994. [16] Z.-J Shang, Volume-preserving maps, source-free and their local structures, J. Phys. A: Math. Gen., 39 (2006), 5601-5615. doi: 10.1088/0305-4470/39/19/S16. [17] Z.-J Shang, "Generating Functions for Volume Preserving Mapping with Applications I: Basic Theory," China/Korea Joint Seminar: Dynamical Systems and Their Application, Available from: http://www.mathnet.or.kr/mathnet/kms_tex/60105.pdf. [18] Z.-J Shang, Construction of volume preserving difference schemes for source-free systems via generating function, Journal of Computational Mathematics, 12 (1994), 265-272. [19] A. Weinstein, The invariance of Poincaré generating function for canonical transformations, Inventiones math., 16 (1972), 202-213. doi: 10.1007/BF01425493. [20] H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411-444. doi: 10.1215/S0012-7094-40-00725-6. [21] H. Xue, O. Verdier and A. Zanna, Discrete Legendre transformation and volume preserving generating forms, In Preparation, (2013). [22] H. Xue and A. Zanna, Explicit volume preserving splitting methods for polynomial divergence-free vector fields, BIT Numerical Mathematics, 53 (2013), 265-281. doi: 10.1007/s10543-012-0394-0.

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References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer, GMT 60, second edition, 1989. [2] M. M. Carroll, A representation theorem for volume preserving transformations, J. International Journal of Non-Linear Mechanics, 39 (2004), 219-224. doi: 10.1016/S0020-7462(02)00167-1. [3] P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems, IMA Journal of Numerical Analysis, 27 (2007), 381-–405. doi: 10.1093/imanum/drl039. [4] K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math, 4 (1986), 279-289. [5] K. Feng and Z.-J Shang, Volume-preserving algorithms for source-free dynamical systems, Numerische Mathematik, 71 (1995), 451-463. doi: 10.1007/s002110050153. [6] K. Feng and D.-L Wang, Dynamical systems and geometric construction of algorithms, Contemporary Mathematics, AMS, Providence, 163 (1994), 1-32. doi: 10.1090/conm/163/01547. [7] K. Feng, H. M. Wu, M.-Z. Qin and D.-L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comp. Math., 7 (1989), 71-96. [8] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations," Springer, Second edition, 2006. [9] A. Iserles, G. R. W. Quispel and P. S. P. Tse, B-series methods cannot be volume-reserving, BIT Numerical Mathematics, 47 (2007), 351-378. doi: 10.1007/s10543-006-0114-8. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [11] H. E. Lomeli and J. D. Meiss, Generating forms for exact volume preserving maps, Discrete and Continuous Dynamical Systems serie S, 2 (2009), 361-377. doi: 10.3934/dcdss.2009.2.361. [12] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [13] J. Moser, "Notes on Dynamical System," Courant Lecture Notes in Mathematics, AMS, New York, 2005. [14] J. Moser and A. P. Veselov, Discrete version of some classical integrable systems and factorization of matrix polynomials, Commun. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [15] J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, 1994. [16] Z.-J Shang, Volume-preserving maps, source-free and their local structures, J. Phys. A: Math. Gen., 39 (2006), 5601-5615. doi: 10.1088/0305-4470/39/19/S16. [17] Z.-J Shang, "Generating Functions for Volume Preserving Mapping with Applications I: Basic Theory," China/Korea Joint Seminar: Dynamical Systems and Their Application, Available from: http://www.mathnet.or.kr/mathnet/kms_tex/60105.pdf. [18] Z.-J Shang, Construction of volume preserving difference schemes for source-free systems via generating function, Journal of Computational Mathematics, 12 (1994), 265-272. [19] A. Weinstein, The invariance of Poincaré generating function for canonical transformations, Inventiones math., 16 (1972), 202-213. doi: 10.1007/BF01425493. [20] H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411-444. doi: 10.1215/S0012-7094-40-00725-6. [21] H. Xue, O. Verdier and A. Zanna, Discrete Legendre transformation and volume preserving generating forms, In Preparation, (2013). [22] H. Xue and A. Zanna, Explicit volume preserving splitting methods for polynomial divergence-free vector fields, BIT Numerical Mathematics, 53 (2013), 265-281. doi: 10.1007/s10543-012-0394-0.
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