TY - JOUR

T1 - Geometric Integration Algorithms on Homogeneous Manifolds

AU - Lewis, Debra

AU - Olver, Peter J.

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2002/11

Y1 - 2002/11

N2 - Given an ordinary differential equation on a homogeneous manifold, one can construct a "geometric integrator" by determining a compatible ordinary differ-ential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is "full," then the order of accuracy of orbit capture (i.e., approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere.

AB - Given an ordinary differential equation on a homogeneous manifold, one can construct a "geometric integrator" by determining a compatible ordinary differ-ential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is "full," then the order of accuracy of orbit capture (i.e., approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere.

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U2 - 10.1007/s102080010028

DO - 10.1007/s102080010028

M3 - Article

AN - SCOPUS:0036435288

VL - 2

SP - 363

EP - 392

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 4

ER -