Stability and $B$-convergence properties of multistep Runge-Kutta methods
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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.References
- W. Auzinger, R. Frank, and G. Kirlinger, A note on convergence concepts for stiff problems, Computing 44 (1990), no.ย 3, 197โ208 (English, with German summary). MR 1058697, DOI 10.1007/BF02262216
- W. Auzinger, R. Frank, and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Numer. Math. 9 (1992), no.ย 2, 91โ109. MR 1147965, DOI 10.1016/0168-9274(92)90008-2
- W. Auzinger, R. Frank, and G. Kirlinger, Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math. 45 (1993), no.ย 1-2, 5โ16. MR 1213581, DOI 10.1016/0377-0427(93)90260-I
- W. Auzinger, R. Frank, and G. Kirlinger, Extending convergence theory for nonlinear stiff problems. I, BIT 36 (1996), no.ย 4, 635โ652. MR 1420268, DOI 10.1007/BF01733784
- K. Burrage, High order algebraically stable multistep Runge-Kutta methods, SIAM J. Numer. Anal. 24 (1987), no.ย 1, 106โ115. MR 874738, DOI 10.1137/0724009
- Kevin Burrage and J. C. Butcher, Nonlinear stability of a general class of differential equation methods, BIT 20 (1980), no.ย 2, 185โ203. MR 583033, DOI 10.1007/BF01933191
- J.C. Butcher, A stability property of implicit Runge-Kutta methods, BIT 15(1975), 358โ361.
- J. C. Butcher, On the implementation of implicit Runge-Kutta methods, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no.ย 3, 237โ240. MR 488746, DOI 10.1007/bf01932265
- Germund Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, pp.ย 60โ72. MR 0448898
- Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, The concept of $B$-convergence, SIAM J. Numer. Anal. 18 (1981), no.ย 5, 753โ780. MR 629662, DOI 10.1137/0718051
- Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no.ย 3, 497โ514. MR 787573, DOI 10.1137/0722030
- Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no.ย 3, 497โ514. MR 787573, DOI 10.1137/0722030
- E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR 1111480, DOI 10.1007/978-3-662-09947-6
- Shoufu Li, $B$-convergence of general linear methods, Proc. BAIL-V International conference, Shanghai, 1988, 203โ208, Boole Press Conf. Ser. 12, 1988.
- Shoufu Li, Stability and $B$-convergence of general linear methods, Proc. 3rd International Congress on Comput. Appl. Math., Belgium, 1988. J. Comput. Appl. Math. 28(1989), 281โ296.
- Shoufu Li, $B$-convergence of General Multivalue Methods Applied to Stiff Problems in Banach Spaces, Proc. National Conf. on Comput. Math., Tianjin, 1990. SCIENCE IN CHINA (Series A), Chinese Series: 1992, 5:476-485, English Series: 36(1993), 1:1-13.
- Shoufu Li, Theory of Computational Methods for Stiff Differential Equations, Science and Technology Press, Hunan, China, 1997.
- Shoufu Li, Multistep Runge-Kutta methods with real eigenvalues and its parallel implementation, Proc. 5-th CSIAM Conference, Qinghua University Publishing House, 1998.
- Shoufu Li, High order algebraically stable and $B$-convergent multistep Runge-Kutta methods with real eigenvalues, To appear.
- Shoufu Li, $B$-convergence properties of multistep Runge-Kutta methods, Research Report, International Conference on Sci. Comput., Hangzhou, 1991. Math. Comput. 62(1994), 565โ575.
- Shoufu Li, Algebraic stability of multistep Runge-Kutta methods, Acta Simulata Systematica Sinica 5(1993), 51โ56 (in Chinese). Also Syst. Engin. Electr., 6(1995), 3:76-82 (in English).
- Josef Schneid, $B$-convergence of Lobatto $\textrm {IIIC}$ formulas, Numer. Math. 51 (1987), no.ย 2, 229โ235. MR 890034, DOI 10.1007/BF01396751
Additional Information
- Shoufu Li
- Affiliation: Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan Province, Peopleโs Republic of China
- Email: lisf@xtu.edu.cn
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1481-1504
- MSC (1991): Primary 65L05; Secondary 65J99
- DOI: https://doi.org/10.1090/S0025-5718-99-01159-X
- MathSciNet review: 1659839