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Numerical solution of isospectral flows

Authors: Mari Paz Calvo, Arieh Iserles and Antonella Zanna
Journal: Math. Comp. 66 (1997), 1461-1486
MSC (1991): Primary 65L05; Secondary 34C30
MathSciNet review: 1434938
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Abstract: In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation

\begin{displaymath}L' = [B(L), L], \quad L(0)=L_0, \end{displaymath}

where $L_0$ is a $d\times d$ symmetric matrix, $B(L)$ is a skew-symmetric matrix function of $L$ and $[B,L]$ is the Lie bracket operator. We show that standard Runge-Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order.

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Additional Information

Mari Paz Calvo
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain

Arieh Iserles
Affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England

Antonella Zanna
Affiliation: Newnham College, University of Cambridge, Cambridge, England

Keywords: Isospectral flows, Runge-Kutta methods, conservation laws, unitary flows, Toda lattice equations.
Received by editor(s): September 7, 1995
Article copyright: © Copyright 1997 American Mathematical Society