Abstract
Two derivative Runge-Kutta methods are Runge-Kutta methods for problems of the form y′ = f(y) that include the second derivative y″ = g(y) = f′(y)f(y) and were developed in the work of Chan and Tsai (Numer. Alg. 53, 171–194 2010). Explicit methods were considered and attention was given to the construction of methods that involve one evaluation of f and many evaluations of g per step. In this work, we consider trigonometrically fitted two derivative explicit Runge-Kutta methods of the general case that use several evaluations of f and g per step; trigonometrically fitting conditions for this general case are given. Attention is given to the construction of methods that involve several evaluations of f and one evaluation of g per step. We modify methods with stages up to four, with three f and one g evaluation and with four f and one g, evaluation based on the fourth and fifth order methods presented in Chan and Tsai (Numer. Alg. 53, 171–194 2010). We provide numerical results to demonstrate the efficiency of the new methods using four test problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Butcher, J. C.: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and general linear methods. Wiley, Chichester (1987)
Chan, R. P. K., Tsai, A.Y.J.: On explicit two-derivative Runge-Kutta methods. Numer. Alg. 53, 171–194 (2010)
Coleman, J.P , Ixaru, L.Gr.: P-stability and exponential fitting methods for y ″ = f(x,y). IMA J. Numer. Anal. 16, 179–199 (1996)
Fang, Y., You, X., Ming, Q.: Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations. Numer. Alg. 65, 651–667 (2014)
Franco, J. M.: Exponentially Fitted explicit Runge Kutta Nyström methods. J. Comput. Appl. Math. 167, 1–19 (2004)
Franco, J. M.: A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187, 41–57 (2006)
Kalogiratou, Z., Monovasilis, Th., Ramos, H., Simos, T.E.: A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)
Kastlunger, K. H., Wanner, G.: Runge Kutta processes with multiple nodes. Computing 9, 9–24 (1972)
Kastlunger, K., Wanner, G.: On Turan type implicit Runge-Kutta methods. Computing 9, 317–325 (1972)
Monovasilis, Th., Kalogiratou, Z., Simos, T.E.: Trigonometrical fitting conditions for two derivative Runge Kutta methods. AIP Conf. Proc. 1790, 150029 (2016)
Sallam, S., Anwar, N.: Sixth order C 2-spline collocation method for integrating second order ordinary initial value problems. Int. J. Comp. Math. 79, 625–635 (2002)
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted explicit Runge-Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000)
Vigo-Aguiar, J., Ramos, H.: Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations. J. Comput. Appl. Math. 158, 187–211 (2003)
Van de Vyver, H.: Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173, 115–130 (2005). 15
Author information
Authors and Affiliations
Corresponding author
Additional information
T. E. Simos is an active member of the European Academy of Sciences and Arts.
Rights and permissions
About this article
Cite this article
Monovasilis, T., Kalogiratou, Z. & Simos, T.E. Trigonometrical fitting conditions for two derivative Runge-Kutta methods. Numer Algor 79, 787–800 (2018). https://doi.org/10.1007/s11075-017-0461-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0461-3