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Stability and $B$-convergence properties
of multistep Runge-Kutta methods

Author: Shoufu Li
Journal: Math. Comp. 69 (2000), 1481-1504
MSC (1991): Primary 65L05; Secondary 65J99
Published electronically: August 17, 1999
MathSciNet review: 1659839
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Abstract: This paper continues earlier work by the same author concerning the stability and $B$-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, $B$- or optimally $B$-convergent are established, by means of which six classes of high order algebraically stable and $B$-convergent multistep Runge-Kutta methods are constructed in a unified pattern. These methods include the class constructed by Burrage in 1987 as special case, and most of them can be regarded as extension of the Gauss, RadauIA, RadauIIA and LobattoIIIC Runge-Kutta methods. We find that the classes of multistep Runge-Kutta methods constructed in the present paper are superior in many respects to the corresponding existing one-step Runge-Kutta schemes.

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Additional Information

Shoufu Li
Affiliation: Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, People’s Republic of China

Keywords: Numerical analysis, nonlinear stability, $B$-convergence, multistep Runge-Kutta methods
Received by editor(s): September 21, 1995
Received by editor(s) in revised form: May 12, 1998, and November 4, 1998
Published electronically: August 17, 1999
Additional Notes: The project supported by National Natural Science Foundation of China.
Article copyright: © Copyright 2000 American Mathematical Society