NDLI: Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

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Computational and Applied Mathematics : Volume 36

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A numerical algorithm for multidimensional modeling of scattered data points

Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation

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Computational and Applied Mathematics : Volume 33

Computational and Applied Mathematics : Volume 32

Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

Content Provider	Springer Nature Link
Author	Dung, La Tri Cong, Nguyen Huu
Copyright Year	2013
Abstract	This paper investigates parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations of the form $$\mathbf{y}''(t)=\mathbf{f}(t,\mathbf{y}(t))$$ . The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of $$s$$ -stage collocation Gauss–Legendre Runge–Kutta–Nyström (RKN) methods based on $$\hat{c}_1,\dots ,\hat{c}_s$$ and $$2s$$ -stage collocation RKN methods based on $$\hat{c}_1,\dots ,\hat{c}_s,1+\hat{c}_1,\dots ,1+\hat{c}_s$$ . At $$n$$ th integration step, the stage values of the $$2s$$ -stage collocation RKN methods evaluated at $$t_n+(1+\hat{c}_1)h,\dots ,t_n+(1+\hat{c}_s)h$$ can be used as the stage values of the collocation Gauss–Legendre RKN method for $$(n+2)$$ th integration step. In this way we obtain the corrector methods in which the integration processes can be proceeded two-step by two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RKN-type (PIRKN-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKNG methods or TBTPIRKNG methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKNG methods to three widely used test problems show that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RKN methods (PIRKN methods) and sequential codes ODEX2 and DOP853 available from the literature.
Starting Page	93
Ending Page	108
Page Count	16
File Format	PDF
ISSN	01018205
Journal	Computational and Applied Mathematics
Volume Number	34
Issue Number	1
e-ISSN	18070302
Language	English
Publisher	Springer Basel
Publisher Date	2014-01-08
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stability and convergence of numerical methods Computational Mathematics and Numerical Analysis Stability Mathematical Applications in Computer Science Runge–Kutta–Nyström methods Predictor–corrector methods Parallelism Initial value problems Applications of Mathematics Mathematical Applications in the Physical Sciences
Content Type	Text
Resource Type	Article

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