Antonella Zanna

Lie-group methods

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

Generalized Polar Decompositions for the Approximation of the Matrix Exponential

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297--324] to approximate a matrix exponential. The algorithms presented have the property that, if

Efficient Computation of the Matrix Exponential by Generalized Polar Decompositions

Z \in {\frak{g}}

The Discrete Moser–Veselov Algorithm for the Free Rigid Body, Revisited

, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of

Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms

{\frak{g}}

Preserving algebraic invariants with Runge-Kutta methods

. This property is very relevant when solving Lie-group ODEs and is not usually fulfilled by standard approximations to the matrix exponential. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. The proposed methods have a complexity of

Adjoint and Selfadjoint Lie-group Methods

{\cal O}(\kappa n^{3})

Runge-Kutta methods for orthogonal and isospectral flows

, with constant

The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

\kappa

Numerical integration of differential equations on homogeneous manifolds

small, depending on the order and the Lie algebra