×

A note on a diagonally implicit Runge-Kutta-Nyström method. (English) Zbl 0637.65065

For the numerical integration of the special second-order initial value problem \(y''=f(t,y),\quad y(t_ 0)=y_ 0,\quad y'(t_ 0)=y_ 0'\) it is often advantageous applying a direct method for this type of differential equations, rather than rewriting that to its first-order form. Fourth-order accurate diagonally implicit (or semi-explicit) Runge- Kutta-Nyström methods with only 2 stages are shown can be obtained. The scheme is given with the largest interval (0,12) of periodicity, and the requirement of P-stability decreases the order to 2.
Reviewer: L.M.Berkovich

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexander, R., Diagonally implicit Runge-Kutta methods for stiff ODEs, SIAM J. Numer. Anal., 14, 1006-1021 (1977) · Zbl 0374.65038
[2] Butcher, J. C., Implicit Runge-Kutta processes, Math. Comp., 18, 50-64 (1964) · Zbl 0123.11701
[3] Hairer, E.; Wanner, G., A theory for Nyström methods, Numer. Math., 25, 383-400 (1976) · Zbl 0307.65053
[4] Hairer, E., Méthodes de Nyström pour l’equation différentielle \(y″ = ƒ(x, y)\), Numer. Math., 27, 283-300 (1977) · Zbl 0325.65033
[5] Hairer, E., Unconditionally stable methods for second order differential equations, Numer. Math., 32, 373-379 (1979) · Zbl 0393.65035
[6] van der Houwen, P. J.; Sommeijer, B. P., Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems, (Report NM-R8704 (1987), Centre for Mathematics and Computer Science: Centre for Mathematics and Computer Science Amsterdam) · Zbl 0676.65072
[7] Lambert, J. D.; Watson, I. A., Symmetric multistep methods for periodic initial value problems, J. Inst. Maths. Applics., 18, 189-202 (1976) · Zbl 0359.65060
[8] Nørsett, S. P., Semi-explicit Runge-Kutta methods, (Report Math. and Comp. No. 6/74 (1974), Dept. of Math., University of Trondheim) · Zbl 0826.65073
[9] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s method, Numer. Math., 13, 154-175 (1969) · Zbl 0219.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.