March  2014, 34(3): 977-990. doi: 10.3934/dcds.2014.34.977

Preserving first integrals with symmetric Lie group methods

1. 

Norwegian University of Science and Technology, Department of Mathematical Sciences, 7491, Trondheim, Norway

Received  January 2013 Revised  May 2013 Published  August 2013

The discrete gradient approach is generalized to yield first integral preserving methods for differential equations in Lie groups.
Citation: Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977
References:
[1]

R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine,, IMA Journal of Numerical Analysis, 22 (2002), 359.  doi: 10.1093/imanum/22.3.359.  Google Scholar

[2]

Luigi Brugnano, Felice Iavernaro and Donato Trigiante., Hamiltonian boundary value methods (energy preserving discrete line integral methods),, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5 (2010), 17.   Google Scholar

[3]

E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical pdes using the "average vector field" method,, Journal of Computational Physics, 231 (2012), 6770.  doi: 10.1016/j.jcp.2012.06.022.  Google Scholar

[4]

E. Celledoni, H. Marthinsen and B. Owren, An introduction to Lie group integrators - basics, new developments and applications,, Journal of Computational Physics, (2013).  doi: 10.1016/j.jcp.2012.12.031.  Google Scholar

[5]

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization,, Acta Numerica, 20 (2011), 1.  doi: 10.1017/S096249291100002X.  Google Scholar

[6]

David Cohen and Ernst Hairer, Linear energy-preserving integrators for Poisson systems,, BIT, 51 (2011), 91.  doi: 10.1007/s10543-011-0310-z.  Google Scholar

[7]

M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs,, SIAM J. Sci. Comput., 33 (2011), 2318.  doi: 10.1137/100810174.  Google Scholar

[8]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[9]

E. Hairer, Energy-preserving variant of collocation methods,, Journal of Numerical Analysis, 5 (2010), 73.   Google Scholar

[10]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215.  doi: 10.1017/S0962492900002154.  Google Scholar

[11]

D. Lewis and J. C. Simo, Nonlinear stability of rotating pseudo-rigid bodies,, Proc. R. Soc. Lond. A, 427 (1990), 281.  doi: 10.1098/rspa.1990.0014.  Google Scholar

[12]

D. Lewis and J. C. Simo, Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups,, J. Nonlinear Sci., 4 (1994), 253.  doi: 10.1007/BF02430634.  Google Scholar

[13]

R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients,, Phil. Trans. Royal Soc. A, 357 (1999), 1021.  doi: 10.1098/rsta.1999.0363.  Google Scholar

[14]

A. Zanna, Collocation and relaxed collocation for the Fer and the Magnus expansions,, SIAM J. Numer. Anal., 36 (1999), 1145.  doi: 10.1137/S0036142997326616.  Google Scholar

show all references

References:
[1]

R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine,, IMA Journal of Numerical Analysis, 22 (2002), 359.  doi: 10.1093/imanum/22.3.359.  Google Scholar

[2]

Luigi Brugnano, Felice Iavernaro and Donato Trigiante., Hamiltonian boundary value methods (energy preserving discrete line integral methods),, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5 (2010), 17.   Google Scholar

[3]

E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical pdes using the "average vector field" method,, Journal of Computational Physics, 231 (2012), 6770.  doi: 10.1016/j.jcp.2012.06.022.  Google Scholar

[4]

E. Celledoni, H. Marthinsen and B. Owren, An introduction to Lie group integrators - basics, new developments and applications,, Journal of Computational Physics, (2013).  doi: 10.1016/j.jcp.2012.12.031.  Google Scholar

[5]

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization,, Acta Numerica, 20 (2011), 1.  doi: 10.1017/S096249291100002X.  Google Scholar

[6]

David Cohen and Ernst Hairer, Linear energy-preserving integrators for Poisson systems,, BIT, 51 (2011), 91.  doi: 10.1007/s10543-011-0310-z.  Google Scholar

[7]

M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs,, SIAM J. Sci. Comput., 33 (2011), 2318.  doi: 10.1137/100810174.  Google Scholar

[8]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[9]

E. Hairer, Energy-preserving variant of collocation methods,, Journal of Numerical Analysis, 5 (2010), 73.   Google Scholar

[10]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215.  doi: 10.1017/S0962492900002154.  Google Scholar

[11]

D. Lewis and J. C. Simo, Nonlinear stability of rotating pseudo-rigid bodies,, Proc. R. Soc. Lond. A, 427 (1990), 281.  doi: 10.1098/rspa.1990.0014.  Google Scholar

[12]

D. Lewis and J. C. Simo, Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups,, J. Nonlinear Sci., 4 (1994), 253.  doi: 10.1007/BF02430634.  Google Scholar

[13]

R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients,, Phil. Trans. Royal Soc. A, 357 (1999), 1021.  doi: 10.1098/rsta.1999.0363.  Google Scholar

[14]

A. Zanna, Collocation and relaxed collocation for the Fer and the Magnus expansions,, SIAM J. Numer. Anal., 36 (1999), 1145.  doi: 10.1137/S0036142997326616.  Google Scholar

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