Abstract
A five-stage fifth-order singly diagonally implicit Runge–Kutta–Nyström method for the integration of second order differential equations possessing an oscillatory solution, is presented in this article. This method is P-stable, which is recommended for problems with a theoretical solution consisting of a periodic part of moderate frequency with a high frequency oscillation with small amplitude superimposed. It also attains an order which is one higher than existing methods of this type. Numerical comparisons with existing methods of this type show its clear advantage.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.M. Chawla and P.S. Rao, High-accuracy P-stable methods for y″ = f(t, y), IMA J. Numer. Anal. 5 (1985) 215–220.
W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961) 381–397.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd ed. (Springer, Berlin/Heidelberg, 1993).
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential–Algebraic Problems, 2nd revised ed. (Springer, Berlin/Heidelberg, 1996).
J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, IMA J. 18 (1976) 189–202.
Matlab, User's Guide (MathWorks Inc., Natick, MA, 1991).
P.W. Sharp, J.M. Fine and K. Burrage, Two-stage and three-stage diagonally implicit Runge–Kutta–Nyström methods of orders three and four, IMA J. Numer. Anal. 10 (1990) 489–504.
E. Stiefel and D.G. Bettis, Stabilization of Cowell's methods, Numer. Math. 13 (1969) 154–175.
P.J. van der Houwen and B.P. Sommeijer, Explicit Runge–Kutta (–Nyström) methods for oscillatory problems, SIAM J. Numer. Anal. 26 (1989) 414–429.
P.J. van der Houwen and B.P. Sommeijer, Diagonally implicit Runge–Kutta–Nyström methods with reduced phase-errors for computing oscillating solutions, SIAM J. Numer. Anal. 26 (1989) 414–429.
P. Van Dooren, Stabilization of Cowell's classical finite difference methods for numerical integration, J. Comput. Phys. 16 (1974) 186–192.
S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, CA, 1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Papageorgiou, G., Famelis, I. & Tsitouras, C. A P-stable singly diagonally implicit Runge–Kutta–Nyström method. Numerical Algorithms 17, 345–353 (1998). https://doi.org/10.1023/A:1016644726305
Issue Date:
DOI: https://doi.org/10.1023/A:1016644726305