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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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 39; no. 1; pp. 128 - 145
Main Authors: Engø, Kenth, Faltinsen, Stig
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2002
Subjects:
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
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
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary: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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SourceType-Scholarly Journals-1
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ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142999364212