Abstract
The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Albrecht,Numerical treatment of O.D.Es.: The theory of A-methods, Numer. Math. 47 (1985), pp. 59–87.
P. Albrecht,A new theoretical approach to Runge-Kutta methods, SIAM J. Numer. Anal. 14 (1987), pp. 391–406.
J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, John Wiley and Sons, New York, 1987.
J. C. Butcher,Diagonally-implicit multi-stage integration methods, Appl. Numer. Math. 11 (1993), pp. 347–364.
J. C. Butcher and Z. Jackiewicz,Diagonally implicit general linear methods for ordinary differential equations, BIT 33 (1993), pp. 452–472.
J. C. Butcher and Z. Jackiewicz,Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations, to appear in Appl. Numer. Math..
J. D. Lambert,Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
Z. Jackiewicz and S. Tracogna,A general class of two-step Runge-Kutta methods for ordinary differential equations, SIAM J. Numer. Anal. 32 (1995), pp. 1390–1427.
Z. Jackiewicz, R. Vermiglio and M. Zennaro,Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations, Appl. Numer. Math. 16 (1995), pp. 343–367.
Author information
Authors and Affiliations
Additional information
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 Consiglio Nazionale delle Richerche.
Rights and permissions
About this article
Cite this article
Jackiewicz, Z., Vermiglio, R. General linear methods with external stages of different orders. Bit Numer Math 36, 688–712 (1996). https://doi.org/10.1007/BF01733788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01733788