Fully parallel Runge-Kutta-Nyström methods for ODEs with oscillating solutions. (English) Zbl 0782.65090

This paper deals with Runge-Kutta-Nyström methods, whose coefficient matrix \(A\) is diagonal. Indeed the algebraic order of these methods is at most two or three but the dispersive order is increasing with the stage number and the methods still remain \(P\)-stable. Two- and three-stage methods are constructed which are very useful in integrating oscillatory problems. Numerical tests are given.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65Y05 Parallel numerical computation
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] Bales, L. A.; Karakashian, O. A.; Serbin, S. M., On the stability of rational approximations to the cosine with only imaginary poles, BIT, 28, 652-658 (1988) · Zbl 0658.65017
[2] Coleman, J. P., Numerical methods for y̋ = f(x,y) via rational approximation for the cosine, IMA J. Numer. Anal., 10, 145-165 (1989) · Zbl 0675.65072
[3] Crisci, M. R., On rational approximation for cos \(z (1991)\), Dipartimento di Matematica e Applicazioni dell’Universitá di Napoli: Dipartimento di Matematica e Applicazioni dell’Universitá di Napoli Napoli, Italy, Preprint No. 48
[4] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, (Springer Series in Computational Mathematics, 8 (1987), Springer: Springer Berlin) · Zbl 1185.65115
[5] Jackson, K. R.; Nørsett, S. P., The potential for parallelism in Runge-Kutta methods, Part 1: RK formulas in standard form, Tech. Rept. No. 239/90 (1990), University of Toronto, Department of Computer Science: University of Toronto, Department of Computer Science Toronto Ont
[6] Jones, W. B.; Thron, W. J., Continued Fractions: Analytic Theory and Applications, (Encyclopedia of Mathematics and Its Applications, Vol II (1980), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0162.09903
[7] Lie, I., Some aspects of parallel Runge-Kutta methods, (Mathematics and Computation Rept. No. 3/87 (1987), NIT: NIT Trondheim, Norway)
[8] Sharp, P. W.; Fine, J. M.; Burrage, K., Two-stage and three-stage diagonally implicit Runge-Kutta-Nyström methods of orders three and four, IMA J. Numer. Anal., 10, 489-504 (1990) · Zbl 0711.65057
[9] van der Houwen, P. J.; Sommejer, B. P., Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solution, SIAM J. Numer. Anal., 24, 595-617 (1987) · Zbl 0624.65058
[10] van der Houwen, P. J.; Sommeijer, B. P., Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems, SIAM J. Numer. Anal., 26, 414-429 (1989) · Zbl 0676.65072
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