Numerical solution of isospectral flows

Calvo, Mari; Iserles, Arieh; Zanna, Antonella

doi:10.1090/S0025-5718-97-00902-2

Numerical solution of isospectral flows
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by Mari Paz Calvo, Arieh Iserles and Antonella Zanna PDF

Math. Comp. 66 (1997), 1461-1486 Request permission

Abstract:

In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation \[ L’ = [B(L), L], \quad L(0)=L_0, \] 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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