Multistep methods integrating ordinary differential equations on manifolds

This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that reflect some of the properties of the multistep methods.

[1]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[2]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[3]  Arne Marthinsen,et al.  Interpolation in Lie Groups , 1999, SIAM J. Numer. Anal..

[4]  H. Munthe-Kaas,et al.  Computations in a free Lie algebra , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  Antonella Zanna.  The method of iterated commutators for ordinary differential equations on Lie groups , 1996 .

[6]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[7]  H. Munthe-Kaas.  Runge-Kutta methods on Lie groups , 1998 .

[8]  P. Crouch,et al.  Numerical integration of ordinary differential equations on manifolds , 1993 .

[9]  A. Zanna,et al.  University of Cambridge Lie-group Methods for Isospectral Flows , 1997 .

[10]  A. Marthinsen,et al.  Modeling and Solution of Some Mechanical Problems on Lie Groups , 1998 .

[11]  Antonella Zanna,et al.  Collocation and Relaxed Collocation for the Fer and the Magnus Expansions , 1999 .

[12]  Arne Marthinsen,et al.  Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames , 1999 .

[13]  R. Carter.  Lie Groups , 1970, Nature.

[14]  Antonella Zanna,et al.  Numerical integration of differential equations on homogeneous manifolds , 1997 .

[15]  H. Munthe-Kaas.  High order Runge-Kutta methods on manifolds , 1999 .

[16]  Antonella Zanna,et al.  Numerical solution of isospectral flows , 1997, Math. Comput..

[17]  Arieh Iserles.  Multistep methods on manifolds , 2001 .

[18]  Arne Marthinsen,et al.  Simulation of ordinary differential equations on manifolds : some numerical experiments and verifications , 1997 .

[19]  E. Gilbert,et al.  Symmetry types of periodic sequences , 1961 .

[20]  Maximal reductions in the Baker–Hausdorff formula , 1993 .

[21]  G. Akrivis.  A First Course In The Numerical Analysis Of Differential Equations [Book News & Reviews] , 1998, IEEE Computational Science and Engineering.

[22]  Arne Marthinsen,et al.  DiffMan: an object-oriented MATLAB toolbox for solving differential equations on manifold , 2001 .

[23]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[24]  J. Oteo.  The Baker-Campbell-Hausdorff formula and nested commutator identities , 1991 .