Abstract.We analyse the attainable order and the stability of Runge-Kutta-Nystri:im (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order of s-stage methods can be raised to 2s for all s. The attainable stage order is one higher and equals s + 1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order. AMS Subject classification: 65M10, 65M20.l. Introduction.In this paper we shall be concerned with the analysis of implicit Runge-KuttaNystrom (RKN methods) based on collocation for integrating the initial-value problem (IVP) for systems of special second-order, ordinary differential equations (ODEs) of dimension d, i.e. the problem,y: IR--. !Rd, f: IR x !Rd--. !Rd, t 0 :::; t :::; T Our motivation for studying implicit RKN methods is the arrival of parallel computers which enables us to solve the implicit relations occurring in the stage vector equation quite efficiently, so that, what is so far considered as the main disadvantage of fully implicit RKN methods, is reduced a great deal. We consider two types of collocation methods for second-order equations: methods based on direct collocation and on indirect collocation (that is, methods obtained by writing the special second-order equation in first-order form and by applying collocation methods for first-order equations [6]). The theory of indirect collocation methods *) These investigations were supported by the University of Amsterdam who provided the third author with a research grant for spending a total of two years in Amsterdam.Received [1]). The main object of the present paper is to extend the work of Kramarz and to derive order and stability results for direct collocation methods. It will be shown that the attainable step point order is similar to that of indirect collocation methods, but the stage order can be raised to s + 1 leaving all but one collocation parameters free.High stage orders are attractive in the case of stiff problems, provided that the method is A or P-stable. However, it seems that the increased-stage-order methods all have finite stability boundaries. If the stage order is decreased to s, then infinite stability boundaries can be obtained. We found A-stable methods with k = s = 2, k = s = 3 and with k = s -1 = 4 implicit stages.We also investigated two stabilizing techniques for achieving A-stability. The first stabilizing technique is based on the preconditioning of the right-hand side in (1.1), that is, stiff components in the right-hand side are damped. In this way, it is possible to transform conditionally stable RKN methods into unconditionally stable preconditioned RKN methods (P RKN methods) at the cost of a slightly more complicated relation for the stage vector. The second stabilizing technique is based on the combination of different, conditionally stable RKN methods. We will give examples of A-stable, composite methods (CRKN methods) with stag...

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Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions. (2000): 65L05, 65L06, 65L20, 65L60. AMS subject classification

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