Abstract
General linear methods are extended to the case in which second derivatives, as well as first derivatives, can be calculated. Methods are constructed of third and fourth order which are A-stable, possess the Runge–Kutta stability property and have a diagonally implicit structure for efficient implementation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.C. Butcher, General linear methods for stiff differential equations, BIT 41 (2001), 240–264.
J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, New York, 2003).
J.C. Butcher and W.M. Wright, The construction of practical general linear methods, BIT 43 (2003) 695–721.
J.R. Cash, Second derivative extended backward differentiation formulas for the numerical integration of stiff systems, SIAM J. Numer. Anal. 18(2) (1981) 21–36.
G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963) 27–43.
G. Hojjati, M.Y. Rahimi Ardabili and S.M. Hosseini, New second derivative multistep methods for stiff systems, Appl. Math. Modelling (to appear).
D. Lee and S. Preiser, A class of nonlinear multistep numerical A-stable methods for solving stiff differential equations, Comput. Math. Appl. 4 (1978) 43–51.
W.M. Wright, General linear methods with inherent Runge–Kutta stability, Ph.D. thesis, The University of Auckland (2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Brezinski
AMS subject classification
65L05
G. Hojjati: Corresponding author.
Rights and permissions
About this article
Cite this article
Butcher, J.C., Hojjati, G. Second derivative methods with RK stability. Numer Algor 40, 415–429 (2005). https://doi.org/10.1007/s11075-005-0413-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11075-005-0413-1