Numer. Math. (2006) 102: 367–381 DOI 10.1007/s00211-005-0627-0 Numerische Mathematik M. P. Calvo · C. Palencia A class of explicit multistep exponential integrators for semilinear problems Received: 5 January 2005 / Published online: 5 December 2005 © Springer-Verlag 2005 Abstract 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 rou- tines 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 1 Introduction In the present paper we derive and analyze a family of explicit multistep exponential methods for the time integration of abstract semilinear problems u  (t) = Au(t) + f (t , u(t )), u(0) = u 0 , 0 ≤ t ≤ T. (1) Problem (1) is assumed to fit in Henry’s setting [6], which covers many inter- esting applications. To be more precise, we assume that A : D(A) ⊂ X → X is the infinitesimal generator of a C 0 -semigroup e tA , t ≥ 0, of linear and bounded operators in a complex Banach space X, with growth governed by e tA ≤ Me ωt , t ≥ 0, (2) for some M> 0, ω ∈ R. The class of nonlinearities allowed in this setting depends on the nature of the semigroup e tA , t ≥ 0. If e tA , t ≥ 0, is just a C 0 -semigroup, we assume, by simplicity, that f :[0,T ]× X → X is globally Lipschitz, i.e. f(t,η) − f(t,ξ)≤ Lη − ξ , η, ξ ∈ X, 0 ≤ t ≤ T, M. P. Calvo (B ) · C. Palencia Departamento de Matemática Aplicada, Universidad de Valladolid, Spain E-mail: maripaz@mac.cie.uva.es