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Numerical solution of isospectral flows |
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| Authors: | Mari Paz Calvo Arieh Iserles Antonella Zanna |
| Institution: | Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain Arieh Iserles ; Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England Antonella Zanna ; Newnham College, University of Cambridge, Cambridge, England |
| Abstract: | In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation
where |
| Keywords: | Isospectral flows Runge-Kutta methods conservation laws unitary flows Toda lattice equations |
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![\begin{displaymath}L' = B(L), L], \quad L(0)=L_0, \end{displaymath}](http://www.ams.org/mcom/1997-66-220/S0025-5718-97-00902-2/gif-abstract/img1.gif)
is a 
symmetric matrix, 
is a skew-symmetric matrix function of 
and ![$B,L]$](http://www.ams.org/mcom/1997-66-220/S0025-5718-97-00902-2/gif-abstract/img6.gif)
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.