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Generating functions for stochastic symplectic methods
Generating functions and volume preserving mappings
1. | Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway, Norway |
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. |
show all references
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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