Adjoint and Selfadjoint Lie-group Methods

In the past few years, a number of Lie-group methods based on Runge—Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge—Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie-group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness.In this paper we discuss Lie-group methods and their dependence on centering coordinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge—Kutta schemes and symmetric coordinates, based on geodesic or on flow midpoint. As result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.

[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]  A. Iserles,et al.  On the solution of linear differential equations in Lie groups , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[4]  Luciano Lopez,et al.  The Cayley transform in the numerical solution of unitary differential systems , 1998, Adv. Comput. Math..

[5]  Kenth Engø-Monsen,et al.  Numerical Integration of Lie-Poisson Systems While Preserving Coadjoint Orbits and Energy , 2001, SIAM J. Numer. Anal..

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

[7]  H. Flaschka.  The Toda lattice. II. Existence of integrals , 1974 .

[8]  A. Bountis.  Dynamical Systems And Numerical Analysis , 1997, IEEE Computational Science and Engineering.

[9]  戸田 盛和.  Theory of nonlinear lattices , 1981 .

[10]  E. Hairer.  Backward analysis of numerical integrators and symplectic methods , 1994 .

[11]  F. Hausdorff,et al.  Die symbolische Exponentialformel in der Gruppentheorie , 2001 .

[12]  Robert I. McLachlan,et al.  On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods , 1995, SIAM J. Sci. Comput..

[13]  A. Zanna.  The Fer expansion and time-symmetry: a Strang-type approach , 2001 .

[14]  S. Faltinsen,et al.  Backward Error Analysis for Lie-Group Methods , 2000 .

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

[16]  A. Perelomov.  The Toda Lattice , 1990 .

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

[18]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[19]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[20]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

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

[22]  Antonella Zanna,et al.  Numerical Integration of Di erential Equations on Homoge-neous Manifolds , 1996 .

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

[24]  Arne Marthinsen,et al.  Norges Teknisk-naturvitenskapelige Universitet Integration Methods Based on Canonical Coordinates of the Second Kind Integration Methods Based on Canonical Coordinates of the Second Kind , 2022 .

[25]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[26]  V. Varadarajan.  Lie groups, Lie algebras, and their representations , 1974 .

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

[28]  R. McLachlan,et al.  Explicit Lie-Poisson integration and the Euler equations. , 1993, Physical review letters.

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

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