Applied Numerical Mathematics 39 (2001) 349–365 www.elsevier.com/locate/apnum Multistep methods integrating ordinary differential equations on manifolds ✩ Stig Faltinsen, Arne Marthinsen ∗ , Hans Z. Munthe-Kaas Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway Abstract This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that reflect some of the properties of the multistep methods.  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Geometric integration; Multistep methods; Numerical integration of ordinary differential equations on manifolds; Numerical analysis; Lie groups; Homogeneous spaces 1. Introduction Classical multistep methods that solve the initial value problem y ′ = f(t,y), y(0) = y 0 ∈ R d , (1) are discussed in a number of texts, see, e.g., [7,8,11]. We write a general k-step method in the form k  i =0 α i y n+i = h k  i =0 β i f(t n+i ,y n+i ), n = 0, 1,..., (2) where α j and β j , j = 0, 1,...,k, are given constants that are independent of the differential equation to be solved, the stepsize h, and n. By rescaling it may be assumed that α k = 1. The method is explicit if β k = 0. ✩ This work was sponsored by The Norwegian Research Council under contract no. 111038/410, through the SYNODE project, http://www.math.ntnu.no/num/synode/. * Corresponding author. E-mail addresses: sf221@damtp.cam.ac.uk (S. Faltinsen), Arne.Marthinsen@math.ntnu.no (A. Marthinsen), Hans.Munthe-Kaas@ii.uib.no (H.Z. Munthe-Kaas). URLs: http://www.damtp.cam.ac.uk/user/na/people.html, http://www.math.ntnu.no/˜arnema/, http://www.ii.uib.no/˜hans/. 0168-9274/01/$20.00  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0168-9274(01)00103-9