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The dependence of Lyapunov exponents of polynomials on their coefficients
Symplectic integration of PDEs using Clebsch variables
1. | School of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, 4442, New Zealand |
2. | Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Gløshaugen, Norway |
Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.
References:
[1] |
L. Brugnano, M. Calvo, J. Montijano and L. Rández, Energy-preserving methods for Poisson systems, Journal of Computational and Applied Mathematics, 236 (2012), 3890–3904, 40 years of numerical analysis: "Is the discrete world an approximation of the continuous one or is it the other way around".
doi: 10.1016/j.cam.2012.02.033. |
[2] |
A. Chern, F. Knöppel, U. Pinkall, P. Schröder and S. Weiẞmann, Schrödinger's smoke, ACM Transactions on Graphics (TOG), 35 (2016), 77.
doi: 10.1145/2897824.2925868. |
[3] |
D. Cohen and E. Hairer,
Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51 (2011), 91-101.
doi: 10.1007/s10543-011-0310-z. |
[4] |
M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 305205, 14pp.
doi: 10.1088/1751-8113/44/30/305205. |
[5] |
D. M. de Diego, Lie-Poisson integrators, preprint, arXiv: 1803.01427, URL https://arXiv.org/abs/1803.01427. Google Scholar |
[6] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[7] |
B. Khesin and R. Wendt, Infinite-dimensional lie groups: Their geometry, orbits, and dynamical systems, The Geometry of Infinite-Dimensional Groups, 2009, 47–153.
doi: 10.1007/978-3-540-77263-7_2. |
[8] |
A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, vol. 53, AMS, 1997.
doi: 10.1090/surv/053. |
[9] |
E. Kuznetsov and A. Mikhailov,
On the topological meaning of canonical Clebsch variables, Physics Letters A, 77 (1980), 37-38.
doi: 10.1016/0375-9601(80)90627-1. |
[10] |
J. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1980), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[11] |
J. E. Marsden and R. Abraham, Foundations of Mechanics, 2nd edition, Addison-Wesley Publishing Co., Redwood City, CA., 1978, URL http://resolver.caltech.edu/CaltechBOOK:1987.001. Google Scholar |
[12] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[13] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer New York, New York, NY, 1999.
doi: 10.1007/978-0-387-21792-5. |
[14] |
R. I. McLachlan,
Spatial discretization of partial differential equations with integrals, IMA Journal of Numerical Analysis, 23 (2003), 645-664.
doi: 10.1093/imanum/23.4.645. |
[15] |
R. I. McLachlan, K. Modin and O. Verdier,
Collective symplectic integrators, Nonlinearity, 27 (2014), 1525-1542.
doi: 10.1088/0951-7715/27/6/1525. |
[16] |
I. Vaisman, Symplectic realizations of poisson manifolds, Lectures on the Geometry of Poisson Manifolds, 1994,115–133.
doi: 10.1007/978-3-0348-8495-2_9. |
[17] |
C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 030, 22 pp.
doi: 10.3842/SIGMA.2008.030. |
show all references
References:
[1] |
L. Brugnano, M. Calvo, J. Montijano and L. Rández, Energy-preserving methods for Poisson systems, Journal of Computational and Applied Mathematics, 236 (2012), 3890–3904, 40 years of numerical analysis: "Is the discrete world an approximation of the continuous one or is it the other way around".
doi: 10.1016/j.cam.2012.02.033. |
[2] |
A. Chern, F. Knöppel, U. Pinkall, P. Schröder and S. Weiẞmann, Schrödinger's smoke, ACM Transactions on Graphics (TOG), 35 (2016), 77.
doi: 10.1145/2897824.2925868. |
[3] |
D. Cohen and E. Hairer,
Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51 (2011), 91-101.
doi: 10.1007/s10543-011-0310-z. |
[4] |
M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 305205, 14pp.
doi: 10.1088/1751-8113/44/30/305205. |
[5] |
D. M. de Diego, Lie-Poisson integrators, preprint, arXiv: 1803.01427, URL https://arXiv.org/abs/1803.01427. Google Scholar |
[6] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[7] |
B. Khesin and R. Wendt, Infinite-dimensional lie groups: Their geometry, orbits, and dynamical systems, The Geometry of Infinite-Dimensional Groups, 2009, 47–153.
doi: 10.1007/978-3-540-77263-7_2. |
[8] |
A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, vol. 53, AMS, 1997.
doi: 10.1090/surv/053. |
[9] |
E. Kuznetsov and A. Mikhailov,
On the topological meaning of canonical Clebsch variables, Physics Letters A, 77 (1980), 37-38.
doi: 10.1016/0375-9601(80)90627-1. |
[10] |
J. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1980), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[11] |
J. E. Marsden and R. Abraham, Foundations of Mechanics, 2nd edition, Addison-Wesley Publishing Co., Redwood City, CA., 1978, URL http://resolver.caltech.edu/CaltechBOOK:1987.001. Google Scholar |
[12] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[13] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer New York, New York, NY, 1999.
doi: 10.1007/978-0-387-21792-5. |
[14] |
R. I. McLachlan,
Spatial discretization of partial differential equations with integrals, IMA Journal of Numerical Analysis, 23 (2003), 645-664.
doi: 10.1093/imanum/23.4.645. |
[15] |
R. I. McLachlan, K. Modin and O. Verdier,
Collective symplectic integrators, Nonlinearity, 27 (2014), 1525-1542.
doi: 10.1088/0951-7715/27/6/1525. |
[16] |
I. Vaisman, Symplectic realizations of poisson manifolds, Lectures on the Geometry of Poisson Manifolds, 1994,115–133.
doi: 10.1007/978-3-0348-8495-2_9. |
[17] |
C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 030, 22 pp.
doi: 10.3842/SIGMA.2008.030. |
























Continuous system | Spatially discretised system |
Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables |
Canonical Hamiltonian ODEs in |
Original PDE, interpreted as a Lie-Poisson equation |
Non-Hamiltonian ODEs in |
Continuous system | Spatially discretised system |
Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables |
Canonical Hamiltonian ODEs in |
Original PDE, interpreted as a Lie-Poisson equation |
Non-Hamiltonian ODEs in |
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