Numerical integration methods for the double-bracket flow.
type: Item
instance of: scholarly article
title: Numerical integration methods for the double-bracket flow.
zbMATH Open document ID: 1085.65057
published in: Journal of Computational and Applied Mathematics
publication date: 2004-03-29T00:00:00Z
review text: This paper is concerned with the numerical solution of the so called double-bracket matrix differential equation \( Y'(t)= [ [ Y(t), N ],] Y(t)], Y(0)= Y_0\) where \(N\) and \(Y_0 \) are \( n \times n \) constant real matrices with \(Y_0\) symmetric, and \( [A,B]= A B-B A\). Since the author is interested in geometric integrators and this equation is a typical example of isospectral flow, the proposed methods retain this property. Taking into account that \( Y(t)\) can be written as \( Y(t)= Q(t) Y_0 Q(t)^T \) with orthogonal \( Q(t)\) and also the representation \( Q(t)= \exp ( \Omega (t))\) with skew-symmetric \( \Omega (t)\) the author proposes a Magnus series expansion up to fourth order of \( \Omega (t)\). This order has been restricted to four to minimize the number of required commutators. A theoretical study of the convergence of the new method is presented that allows the author to include dependence on the matrix norm selected. Finally, some numerical experiments comparing the new method with other fourth order Lie-group Runge-Kutta (RK) integrator due to \textit{H. Munthe-Kaas} and \textit{B. Owren} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1754, 957--981 (1999; Zbl 0956.65056)] which also preserves isospectrality and the classical RK4 method of order four are presented. As expected geometric integrators provide much better approximations to the eigenvalues than the classical RK4. Moreover the new method shows much better stability properties than the RK of Munthe-Kaas and Owren [loc. cit.].
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 2062748
zbMATH Keywords: geometric integration, Magnus expansions
zbMATH Keywords: Lie-group solvers
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: double-bracket matrix differential equation
zbMATH Keywords: isospectral flow
zbMATH Keywords: convergence
zbMATH Keywords: numerical experiments
zbMATH Keywords: stability
author: Fernando Casas
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: j.cam.2003.08.046
OpenAlex ID: W2124378605
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DOI: 10.1016/J.CAM.2003.08.046
description: scientific article; zbMATH DE number 2062748