Numerical Integration of Lie-Poisson Systems while Preserving Coadjoint Orbits and Energy
In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie-Poisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra$\mathprak{g}^\ast$to advance the numerical flow, we devise methods of arbitrary order that automatical...
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| Veröffentlicht in: | SIAM journal on numerical analysis Jg. 39; H. 1; S. 128 - 145 |
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Society for Industrial and Applied Mathematics
2002
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| Abstract | In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie-Poisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra$\mathprak{g}^\ast$to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body and a finite-dimensional truncation of the Euler equations for a two-dimensional (2D) incompressible fluid are used to illustrate the properties of the algorithm. |
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| AbstractList | In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie--Poisson systems numerically. By using the coadjoint action of the Lie group $G$ on the dual Lie algebra ${\mbox{\normalsize$\mathfrak{g}$}}^*$ to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body and a finite-dimensional truncation of the Euler equations for a two-dimensional (2D) incompressible fluid are used to illustrate the properties of the algorithm. In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie-Poisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra$\mathprak{g}^\ast$to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body and a finite-dimensional truncation of the Euler equations for a two-dimensional (2D) incompressible fluid are used to illustrate the properties of the algorithm. |
| Author | Faltinsen, Stig Engø, Kenth |
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| Cites_doi | 10.1103/PhysRevLett.71.3043 10.1006/jcph.1993.1128 10.1016/0375-9601(88)90773-6 10.1007/BF01003559 10.1090/S0025-5718-00-01223-0 10.1098/rsta.1999.0363 10.1002/cpa.3160470505 10.1137/0916010 10.1007/978-1-4612-1126-6 10.1023/A:1022362117414 10.1007/BF02352494 10.1016/0167-2789(91)90152-Y 10.1137/S1064827595285494 10.1007/b97593 10.1023/A:1018908700358 10.1007/978-1-4899-3093-4 10.1016/0167-2789(91)90081-J 10.1137/S0036142997329797 10.1007/978-0-387-21792-5 10.1007/s003329900018 10.1063/1.532892 10.1088/0951-7715/12/6/314 10.1016/0167-2789(94)90046-9 10.1007/BF01212956 10.1007/BF02430634 10.1098/rsta.1999.0360 10.1023/A:1021950708869 10.1103/PhysRevE.48.3643 10.1023/A:1022336301001 |
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| Keywords | Euler equation Action Error estimation Iteration Machine System Truncation Time dependence Conservation law Numerical computation Energy Coadjoint orbit Orbit determination Integrator Lie Poisson equation Skew symmetric matrix Growth of error Rigid bodies Dimensionality Numerical integration Orbit Poisson equation Algorithm Flow Two dimensional equation Hamilton equation First integral Linear algebra Hamiltonian system Lie group Incompressible fluid Lie algebra |
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| Snippet | In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie-Poisson systems numerically. By using the coadjoint action of... In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie--Poisson systems numerically. By using the coadjoint action of... |
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| SubjectTerms | Algebra Algorithms Applied mathematics Classical and quantum physics: mechanics and fields Classical mechanics of discrete systems: general mathematical aspects Coordinate systems Differential equations Energy Exact sciences and technology Lie groups Mathematical analysis Mathematical integration Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods Orbits Ordinary differential equations Physics Rigid structures Sciences and techniques of general use Trapezoidal rule Truncation |
| Title | Numerical Integration of Lie-Poisson Systems while Preserving Coadjoint Orbits and Energy |
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