Abstract
A general class of variable stepsize continuous two-step Runge-Kutta methods is investigated. These methods depend on stage values at two consecutive steps. The general convergence and order criteria are derived and examples of methods of orderp and stage orderq=p orq=p−1 are given forp≤5. Numerical examples are presented which demonstrate that high order and high stage order are preserved on nonuniform meshes with large variations in ratios between consecutive stepsizes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Albrecht, Numerical treatment of ODEs: the theory ofA-methods, Numer. Math. 47 (1985) 59–87.
P. Albrecht, A new theoretical approach to Runge-Kutta methods, SIAM J. Numer. Anal. 24 (1987) 391–406.
P. Albrecht, Elements of a general theory of composite integration methods, Appl. Math. Comp. 31 (1989) 1–17.
J.C. Butcher, Diagonally implicit multi-stage integration methods, Appl. Numer. Math. 11 (1993) 347–363.
C.W. Gear, The effect of variable mesh size on the stability of multistep methods, SIAM J. Numer. Anal. 11 (1974) 1025–1043.
T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972) 603–637.
Z. Jackiewicz, R. Renaut and M. Zennaro, Explicit two-step Runge-Kutta methods, Appl. Math. 40 (1995) 433–456.
Z. Jackiewicz and S. Tracogna, A general class of two-step Runge-Kutta methods for ordinary differential equations, SIAM J. Numer. Anal. 32 (1995) 1390–1427.
Z. Jackiewicz and M. Zennaro, Variable-stepsize explicit two-step Runge-Kutta methods, Math. Comp. 59 (1992) 421–438.
S. Tracogna, Implementation of two-step Runge-Kutta methods for ordinary differential equations, submitted.
Author information
Authors and Affiliations
Additional information
Communicated by J.C. Butcher
The work of the first author was supported by the National Science Foundation under grant NSF DMS-9208048. The work of the second author was supported by the Italian Government.
Rights and permissions
About this article
Cite this article
Jackiewicz, Z., Tracogna, S. Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations. Numer Algor 12, 347–368 (1996). https://doi.org/10.1007/BF02142812
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02142812