Numer. Math. 38, 193-208(1981) Numerische Mathematik 9 Springer-Verlag 1981 Perturbed Collocation and Runge-Kutta Methods S.P. Norsett I and G. Wanner 2 a Institute of Numerical Mathematics NTH, N-7034Trondheim, Norway 2 Universit6 de Gen6ve, Section de Math6matiques, Case postale 124, CH-1211 Gen6ve 24, Switzerland Summary. It is well known that some implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order and A,B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural ques- tion, if all RK methods can be considered as a somewhat "perturbed" collocation. After having introduced this notion we give a proof on the order of the method and discuss their stability properties. Much of known theory becomes simple and beautiful. Subject Classifications: AMS(MOS): 65L05; Cr: 5.17. 1. Perturbed Collocation We consider the numerical solution of y'=f(t,y), y(to)=Yo, y~IR". (1) Given h >0 we want to find an approximation u(t) to the exact solution y(t) for to <=t<=to+h. Definition 1. Let //m he the linear space of polynomials of degree <m and Nfillm 1 m ~(t) =7., ,_Z ~ (p,~-,~,~) t' j = 1 .... ,m (2) be given polynomials. Then we define the perturbation operator Pto,h:I-Im--~Mm by (Pro, h u)(t) = u(t) + ~, Nj((t -- to)/h ) u~ ) h j (3) j=l where u0 ~) denotes the derivative uO)(to). This definition is "natural" in the sense that it commutes with affine transformations of t as follows: 0029-599X/81/0038/0193/$03.20