April  2016, 36(4): 2285-2303. doi: 10.3934/dcds.2016.36.2285

A classification of volume preserving generating forms in $\mathbb{R}^3$

1. 

Department of Computing, Mathematics and Physics, Bergen University College, 5063 Bergen, Norway

2. 

Department of Mathematics, University of Bergen, 5020 Bergen, Norway

3. 

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

Received  December 2014 Revised  July 2015 Published  September 2015

In earlier work, Lomeli and Meiss [9] used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. In [20], Xue and Zanna studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators. In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction. We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in $\mathbb{R}^3$.
Citation: Olivier Verdier, Huiyan Xue, Antonella Zanna. A classification of volume preserving generating forms in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2285-2303. doi: 10.3934/dcds.2016.36.2285
References:
[1]

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.

[2]

K. Feng, Difference schemes for hamiltonian formalism and symplectic geometry, J. Comput. Math, 4 (1986), 279-289.

[3]

K. Feng, H.M. Wu, M.-Z. Qin and D.-L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 7 (1989), 71-96.

[4]

K. Feng and Z. J. Shang, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463. doi: 10.1007/s002110050153.

[5]

H. Goldstein, C. P. P. Jr. and J. L. Safko, Classical Mechanics, 3rd edition, Pearson, 2001. doi: 10.1063/1.3067728.

[6]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer series in computational mathematics, Springer, 2006.

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edition, Springer-Verlag, Berlin, 1993.

[8]

A. Iserles, G. R. W. Quispel and P. S. P. Tse, B-series methods cannot be volume-preserving, BIT, 47 (2007), 351-378. doi: 10.1007/s10543-006-0114-8.

[9]

H. E. Lomelí and J. D. Meiss, Generating forms for exact volume-preserving maps, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 361-377. doi: 10.3934/dcdss.2009.2.361.

[10]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[11]

R. I. McLachlan, H. Z. Munthe-Kaas, G. R. W. Quispel and A. Zanna, Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Found. Comput. Math., 8 (2008), 335-355. doi: 10.1007/s10208-007-9009-6.

[12]

R. McLachlan and G. Quispel, Explicit geometric integration of polynomial vector fields, BIT Numerical Mathematics, 44 (2004), 515-538. doi: 10.1023/B:BITN.0000046814.29690.62.

[13]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053.

[14]

G. R. W. Quispel, Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30. doi: 10.1016/0375-9601(95)00586-R.

[15]

G. Quispel and D. McLaren, Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows, J. Comp. Phys., 186 (2003), 308-316. doi: 10.1016/S0021-9991(03)00068-8.

[16]

Z. J. Shang, Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.

[17]

Z. Shang, Generating functions for volume-preserving mappings and Hamilton-Jacobi equations for source-free dynamical systems, Sci. China, Ser. A, 37 (1994), 1172-1188.

[18]

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.

[19]

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.

[20]

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

[21]

A. Zanna, Explicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions, IMA J. Numer. Anal., 35 (2015), 89-106. doi: 10.1093/imanum/drt070.

show all references

References:
[1]

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.

[2]

K. Feng, Difference schemes for hamiltonian formalism and symplectic geometry, J. Comput. Math, 4 (1986), 279-289.

[3]

K. Feng, H.M. Wu, M.-Z. Qin and D.-L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 7 (1989), 71-96.

[4]

K. Feng and Z. J. Shang, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463. doi: 10.1007/s002110050153.

[5]

H. Goldstein, C. P. P. Jr. and J. L. Safko, Classical Mechanics, 3rd edition, Pearson, 2001. doi: 10.1063/1.3067728.

[6]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer series in computational mathematics, Springer, 2006.

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edition, Springer-Verlag, Berlin, 1993.

[8]

A. Iserles, G. R. W. Quispel and P. S. P. Tse, B-series methods cannot be volume-preserving, BIT, 47 (2007), 351-378. doi: 10.1007/s10543-006-0114-8.

[9]

H. E. Lomelí and J. D. Meiss, Generating forms for exact volume-preserving maps, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 361-377. doi: 10.3934/dcdss.2009.2.361.

[10]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[11]

R. I. McLachlan, H. Z. Munthe-Kaas, G. R. W. Quispel and A. Zanna, Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Found. Comput. Math., 8 (2008), 335-355. doi: 10.1007/s10208-007-9009-6.

[12]

R. McLachlan and G. Quispel, Explicit geometric integration of polynomial vector fields, BIT Numerical Mathematics, 44 (2004), 515-538. doi: 10.1023/B:BITN.0000046814.29690.62.

[13]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053.

[14]

G. R. W. Quispel, Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30. doi: 10.1016/0375-9601(95)00586-R.

[15]

G. Quispel and D. McLaren, Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows, J. Comp. Phys., 186 (2003), 308-316. doi: 10.1016/S0021-9991(03)00068-8.

[16]

Z. J. Shang, Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.

[17]

Z. Shang, Generating functions for volume-preserving mappings and Hamilton-Jacobi equations for source-free dynamical systems, Sci. China, Ser. A, 37 (1994), 1172-1188.

[18]

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.

[19]

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.

[20]

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

[21]

A. Zanna, Explicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions, IMA J. Numer. Anal., 35 (2015), 89-106. doi: 10.1093/imanum/drt070.

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