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 & Continuous Dynamical Systems - A, 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.  doi: 10.1093/imanum/drl039.  Google Scholar

[2]

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

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems,, 2nd edition, (1993).   Google Scholar

[8]

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

[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.  doi: 10.3934/dcdss.2009.2.361.  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]

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.  doi: 10.1007/s10208-007-9009-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

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.  doi: 10.1093/imanum/drl039.  Google Scholar

[2]

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

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems,, 2nd edition, (1993).   Google Scholar

[8]

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

[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.  doi: 10.3934/dcdss.2009.2.361.  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]

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.  doi: 10.1007/s10208-007-9009-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

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