Norges Teknisk-naturvitenskapelige Universitet Integration Methods Based on Canonical Coordinates of the Second Kind Integration Methods Based on Canonical Coordinates of the Second Kind

Summary. We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

[1]  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.

[2]  M. V. Leeuwen,et al.  LiE, a software package for Lie group computations , 1994 .

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

[4]  Elena Celledoni,et al.  Approximating the exponential from a Lie algebra to a Lie group , 2000, Math. Comput..

[6]  Arne Marthinsen,et al.  Multistep methods integrating ordinary differential equations on manifolds , 2001 .

[7]  Feng Kang,et al.  Volume-preserving algorithms for source-free dynamical systems , 1995 .

[8]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

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

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

[11]  A. Iserles,et al.  Methods for the approximation of the matrix exponential in a Lie‐algebraic setting , 1999, math/9904122.

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

[13]  I. G. MacDonald,et al.  Lectures on Lie groups and Lie algebras , 1995 .

[14]  Kenth Engø-Monsen,et al.  On the construction of geometric integrators in the RKMK class , 2000 .

[15]  Kenth Engø.  On the Construction of Geometric Integrators in the RKMK Class , 2000 .

[16]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[17]  W. Ledermann.  INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY , 1974 .

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