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Burrage, Kevin; Butcher, J. C.

Non-linear stability of a general class of differential equation methods. (English) Zbl 0431.65051

BIT, Nord. Tidskr. Inf.-behandl. 20, 185-203 (1980).
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Cited in 11 Reviews
Cited in 121 Documents

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations

Keywords:

stability analysis; nonlinear system of differential equations; multistep methods; multileg methods; Runge-Kutta methods; contractivity condition; G-stability; monotonic methods; algebraic stability; coercivity condition; study of error growth; non-linear stability analysis; error propagation
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\textit{K. Burrage} and \textit{J. C. Butcher}, BIT, Nord. Tidskr. Inf.-behandl. 20, 185--203 (1980; Zbl 0431.65051)
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References:

[1] K. Burrage and J. C. Butcher,Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979), 46–57. · Zbl 0396.65043 · doi:10.1137/0716004
[2] K. Burrage,High order algebraically stable Runge-Kutta methods, BIT 18 (1978), 373–383. · Zbl 0401.65049 · doi:10.1007/BF01932017
[3] J. C. Butcher,On the convergence of numerical solutions to ordinary differential equations, Math. Comp. 20 (1966), 1–10. · Zbl 0141.13504 · doi:10.1090/S0025-5718-1966-0189251-X
[4] J. C. Butcher,The order of numerical methods for ordinary differential equations, Math. Comp. 27 (1973), 793–806. · Zbl 0278.65073 · doi:10.1090/S0025-5718-1973-0343620-7
[5] J. C. Butcher,A stability property of implicit Runge-Kutta methods, BIT 15 (1975), 358–361. · Zbl 0333.65031 · doi:10.1007/BF01931672
[6] G. J. Cooper,The order of convergence of general linear methods for ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), 643–661. · Zbl 0398.65040 · doi:10.1137/0715043
[7] M. Crouzeix,Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979), 75–82. · Zbl 0431.65052 · doi:10.1007/BF01397651
[8] G. Dahlquist,A special stability problem for linear multistep methods, BIT 3 (1963), 27–43. · Zbl 0123.11703 · doi:10.1007/BF01963532
[9] G. Dahlquist,On stability and error analysis for stiff non-linear problems, 1, Dept. of Information Processing, Royal Institute of Technology, Stockholm, Report NA 75.08.
[10] G. Dahlquist,G-stability is equivalent to A-stability, BIT 18 (1978), 384–401. · Zbl 0413.65057 · doi:10.1007/BF01932018
[11] C. W. Gear,Numerical initial value problems in ordinary differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1971. · Zbl 1145.65316
[12] E. Hairer and G. Wanner,Multistep-multistage-multiderivative methods for ordinary differential equations, Computing 11 (1973), 287–303. · Zbl 0271.65048 · doi:10.1007/BF02252917
[13] E. Hairer and G. Wanner,On the Butcher group and general multi-value methods, Computing 13 (1974), 1–15. · Zbl 0293.65050 · doi:10.1007/BF02268387
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