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A class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided. Mathematics Subject Classifications (2000)65J15 · 65M12 · 65L05 · 65M20

Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equationWe 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.1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential equationwhereThe choice of the matrix function B(L) characterizes the dynamics of the underlying flow L(t). Important special cases are the Toda lattice equations, doublebracket flows and KvM flows.Toda lattice equations in the Lax formulation (1)