Two-step Runge-Kutta Methods with Quadratic Stability Functions

“…Following the above described lines drawn in the literature in the context of GLMs for first order ODEs (also compare [28][29][30][31]), we introduce an analogous notion of stability for GLN methods (1.2), in order to let these methods inherit the same stability properties of a certain RKN method assumed as a reference.…”

Section: Runge-kutta-nyström Stabilitymentioning

confidence: 99%

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

D’Ambrosio

Paternoster

2014

Journal of Computational and Applied Mathematics

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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.

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“…Results in this section are based on those obtained by Conte, D'Ambrosio and Jackiewicz [12]. To investigate the Nordsieck methods with so called quadratic stability function, i.e stability function with two nonzero roots, we introduce equivalence relation between matrices of the same dimensions.…”

Section: Criteria For Quadratic Stabilitymentioning

confidence: 99%

“…In Section 3 we present some conditions which guarantee that stability function has only two nonzero roots. These results are based on work of D'Ambrosio, Izzo and Jackiewicz [12] for two-step Runge-Kutta methods. In Sections 4, 5 and 6 we use these criteria to construct methods up to order 4.…”

Section: Introductionmentioning

confidence: 99%