Generating functions and volume preserving mappings

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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

Abstract
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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.

Keywords: differential forms., symplectic, Generating function, volume preserving.

Mathematics Subject Classification: Primary: 53D22, 70H15; Secondary: 65L0.

Citation: Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229

References:

[1]	V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Springer, (1989). Google Scholar
[2]	M. M. Carroll, A representation theorem for volume preserving transformations,, J. International Journal of Non-Linear Mechanics, 39 (2004), 219. doi: 10.1016/S0020-7462(02)00167-1. Google Scholar
[3]	P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems,, IMA Journal of Numerical Analysis, 27 (2007). doi: 10.1093/imanum/drl039. Google Scholar
[4]	K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry,, J. Comput. Math, 4 (1986), 279. Google Scholar
[5]	K. Feng and Z.-J Shang, Volume-preserving algorithms for source-free dynamical systems,, Numerische Mathematik, 71 (1995), 451. doi: 10.1007/s002110050153. Google Scholar
[6]	K. Feng and D.-L Wang, Dynamical systems and geometric construction of algorithms,, Contemporary Mathematics, 163 (1994), 1. doi: 10.1090/conm/163/01547. Google Scholar
[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. Google Scholar
[8]	E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", Springer, (2006). Google Scholar
[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. doi: 10.1007/s10543-006-0114-8. Google Scholar
[10]	J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[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. doi: 10.3934/dcdss.2009.2.361. Google Scholar
[12]	R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numerica, 11 (2002), 341. doi: 10.1017/S0962492902000053. Google Scholar
[13]	J. Moser, "Notes on Dynamical System,", Courant Lecture Notes in Mathematics, (2005). Google Scholar
[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. doi: 10.1007/BF02352494. Google Scholar
[15]	J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems,", Chapman & Hall, (1994). Google Scholar
[16]	Z.-J Shang, Volume-preserving maps, source-free and their local structures,, J. Phys. A: Math. Gen., 39 (2006), 5601. doi: 10.1088/0305-4470/39/19/S16. Google Scholar
[17]	Z.-J Shang, "Generating Functions for Volume Preserving Mapping with Applications I: Basic Theory,", China/Korea Joint Seminar: Dynamical Systems and Their Application, (). Google Scholar
[18]	Z.-J Shang, Construction of volume preserving difference schemes for source-free systems via generating function,, Journal of Computational Mathematics, 12 (1994), 265. Google Scholar
[19]	A. Weinstein, The invariance of Poincaré generating function for canonical transformations,, Inventiones math., 16 (1972), 202. doi: 10.1007/BF01425493. Google Scholar
[20]	H. Weyl, The method of orthogonal projection in potential theory,, Duke Math. J., 7 (1940), 411. doi: 10.1215/S0012-7094-40-00725-6. Google Scholar
[21]	H. Xue, O. Verdier and A. Zanna, Discrete Legendre transformation and volume preserving generating forms,, In Preparation, (2013). Google Scholar
[22]	H. Xue and A. Zanna, Explicit volume preserving splitting methods for polynomial divergence-free vector fields,, BIT Numerical Mathematics, 53 (2013), 265. doi: 10.1007/s10543-012-0394-0. Google Scholar

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References:

[1]	V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Springer, (1989). Google Scholar
[2]	M. M. Carroll, A representation theorem for volume preserving transformations,, J. International Journal of Non-Linear Mechanics, 39 (2004), 219. doi: 10.1016/S0020-7462(02)00167-1. Google Scholar
[3]	P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems,, IMA Journal of Numerical Analysis, 27 (2007). doi: 10.1093/imanum/drl039. Google Scholar
[4]	K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry,, J. Comput. Math, 4 (1986), 279. Google Scholar
[5]	K. Feng and Z.-J Shang, Volume-preserving algorithms for source-free dynamical systems,, Numerische Mathematik, 71 (1995), 451. doi: 10.1007/s002110050153. Google Scholar
[6]	K. Feng and D.-L Wang, Dynamical systems and geometric construction of algorithms,, Contemporary Mathematics, 163 (1994), 1. doi: 10.1090/conm/163/01547. Google Scholar
[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. Google Scholar
[8]	E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", Springer, (2006). Google Scholar
[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. doi: 10.1007/s10543-006-0114-8. Google Scholar
[10]	J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[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. doi: 10.3934/dcdss.2009.2.361. Google Scholar
[12]	R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numerica, 11 (2002), 341. doi: 10.1017/S0962492902000053. Google Scholar
[13]	J. Moser, "Notes on Dynamical System,", Courant Lecture Notes in Mathematics, (2005). Google Scholar
[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. doi: 10.1007/BF02352494. Google Scholar
[15]	J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems,", Chapman & Hall, (1994). Google Scholar
[16]	Z.-J Shang, Volume-preserving maps, source-free and their local structures,, J. Phys. A: Math. Gen., 39 (2006), 5601. doi: 10.1088/0305-4470/39/19/S16. Google Scholar
[17]	Z.-J Shang, "Generating Functions for Volume Preserving Mapping with Applications I: Basic Theory,", China/Korea Joint Seminar: Dynamical Systems and Their Application, (). Google Scholar
[18]	Z.-J Shang, Construction of volume preserving difference schemes for source-free systems via generating function,, Journal of Computational Mathematics, 12 (1994), 265. Google Scholar
[19]	A. Weinstein, The invariance of Poincaré generating function for canonical transformations,, Inventiones math., 16 (1972), 202. doi: 10.1007/BF01425493. Google Scholar
[20]	H. Weyl, The method of orthogonal projection in potential theory,, Duke Math. J., 7 (1940), 411. doi: 10.1215/S0012-7094-40-00725-6. Google Scholar
[21]	H. Xue, O. Verdier and A. Zanna, Discrete Legendre transformation and volume preserving generating forms,, In Preparation, (2013). Google Scholar
[22]	H. Xue and A. Zanna, Explicit volume preserving splitting methods for polynomial divergence-free vector fields,, BIT Numerical Mathematics, 53 (2013), 265. doi: 10.1007/s10543-012-0394-0. Google Scholar

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