P-stable general Nyström methods fory″=f(y(t

Abstract

a b s t r a c tWe focus our attention on the family of General Linear Methods (GLMs), for the numerical solution of second order ordinary differential equations (ODEs). These are multivalue methods introduced in (D' Ambrosio et al., 2012) [3] with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order ODEs. Our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of GLMs for second order ODEs: thus, we present the extension of the classical notions of stability matrix, stability polynomial, stability and periodicity interval, A-stability and P-stability to the family of GLMs. Special attention will be given to the derivation of highly stable GLMs, whose stability properties depend on the stability polynomial of indirect Runge-Kutta-Nyström methods based on Gauss-Legendre collocation points, which are known to be P-stable. In this way, we are able to provide P-stable GLMs whose order of convergence is greater than that of the corresponding RKN method, without heightening the computational cost. We finally provide and discuss examples of P-stable irreducible GLMs satisfying the mentioned features.

General Nyström methods in Nordsieck form: Error analysis

“…In addition, in order to exploit the generality of the approach and the large number of degrees of freedom involved in the formulation of the methods, we will also aim to treat constructive issues, even by means of optimization techniques, in order to derive new examples of methods which improve existing ones. Up to now, the author have derived a first example in [9] of methods which results more accurate and efficient than the analog Runge-Kutta-Nyström.…”

“…In particular, we have assumed that the methods provide approximations to the Nordsieck vector (1.6) at each step point. For such methods, some issues regarding order conditions and convergence have been given by employing the theory of rooted trees, leading to general order conditions up to order 4 that are satisfied in a more general setting than that described in [9]: indeed, the order conditions provided in [9] hold true only for high stage order methods; this hypothesis is here neglected. In addition, an accuracy analysis has been provided, in order to obtain a possible representation of the local truncation error associated to (1.2).…”

“…If q = p or q = p − 1, all the terms up to order O(h p ) vanish (compare [9]), and the local truncation error takes the form …”

“…We focus our attention on the numerical solution of special second order Ordinary Differential Equations (ODEs) introduced in [8] and furtherly investigated in [9,10,11], as extension of the family of General Linear Methods for first order ODEs [1,5,6,18]. The procedures involved in .…”

Order conditions for general linear methods

“…Such high order and stage order methods with some desirable stability properties (large regions of absolute stability for explicit methods, A-, L-, and algebraic stability for implicit methods) were investigated in a series of papers [29,56,30,[16][17][18][19][20][21][22][23][24][25][26][27]33,34,[57][58][59] and the monograph [3]. In the next section we will derive the general order conditions for GLMs (1.5) without restrictions on stage order (except stage preconsistency and stage consistency conditions (3.11) and (3.12)) using the approach proposed by Albrecht [60][61][62][63][64] in the context of RK, composite and linear cyclic methods, and generalized by Jackiewicz and Tracogna [39,40] and Tracogna [43], to TSRK methods for ODEs, by Jackiewicz and Vermiglio [65] to general linear methods with external stages of different orders, and by Garrappa [66] to some classes of Runge-Kutta methods for Volterra integral equations with weakly singular kernels.…”