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Zero-stability properties of the three-ordinate variable stepsize variable formula methods. (English) Zbl 0475.65041


MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

Software:

DIFSUB

References:

[1] Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand.4, 33–53 (1956) · Zbl 0071.11803
[2] Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Tech., Stockholm, No. 130 1959 · Zbl 0085.33401
[3] Dahlquist, G.: Error analysis for a class of methods of stiff non-linear initial value problems. In: Numerical analysis. Proceedings of the Dundee Conference on Numerical Analysis. (G. Watson, ed.) Lecture Notes in Mathematics, Vol. 506, pp. 60–74. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0352.65042
[4] Dahlquist, G.:G-stability is equivalent toA-stability. BIT18, 384–401 (1978) · Zbl 0413.65057 · doi:10.1007/BF01932018
[5] Dahlquist, G.: Some properties of linear multistep and one-leg methods for ordinary differential equations. In: Working papers for the 1979 Signum Meeting on Numerical Ordinary Differential Equations. Department of Computer Science, University of Illionis at Urbana-Champaign, Urbana, USA, 1979 · Zbl 0489.65044
[6] Gear, C.W.: Stability of variable-step methods for ordinary differential equations. Techn. Report, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA, 1978 · Zbl 0388.65030
[7] Gear, C.W., Tu, K.W.: The effects of variable mesh size on the stability of multistep methods. SIAM J. Numer. Anal.11, 1025–1043 (1974) · Zbl 0292.65041 · doi:10.1137/0711079
[8] Gear, C.W., Watanabe, D.S.: Stability and convergence of variable order multistep methods. SIAM J. Numer. Anal.11, 1044–1058 (1974) · Zbl 0294.65041 · doi:10.1137/0711080
[9] Henrici, P.: Descrete variable methods in ordinary differential equations. New York, London: J. Wiley, 1962 · Zbl 0112.34901
[10] Jeltsch, R., Nevanlinna, O.: Stability of explicit time discretizations for solving initial value problems. Techn. Report No. 30, Mathematics, University of Oulu, Linnanmaa, Oulu, Finland, 1979 · Zbl 0457.65054
[11] Lambert, J.D.: Computational methods in ordinary differential equations. New York, London: J. Wiley 1973 · Zbl 0258.65069
[12] Piotrowski, P.: Stability, consistency and convergence of variableK-step methods for numerical integration of large systems of ordinary differential equations. In: Conference on numerical solution of differential equations (J.L. Morris, ed.), pp. 221–227. Berlin-Heidelberg-New York: Springer 1969
[13] Shampine, L.F., Gordon, M.K.: Computer solution of ordinary differential equations: The initial value problem. San Francisco: Freeman 1975 · Zbl 0347.65001
[14] Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0276.65001
[15] Thomsen, P.G., Zlatev, Z.: Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations. BIT 19, 503–517 (1979) · Zbl 0428.65047 · doi:10.1007/BF01931267
[16] Zlatev, Z.: Stability properties of variable stepsize variable formula methods. Numer. Math.31, 175–182 (1978) · doi:10.1007/BF01397474
[17] Zlatev, Z., Thomsen, P.G.: Application of backward differentiation methods to the finite element solution of time-dependent problems. Internat. J. Numer. Methods Engrg.14, 1051–1061 (1979) · Zbl 0416.65054 · doi:10.1002/nme.1620140708
[18] Zlatev, Z., Thomsen, P.G.: Automatic solution of differential equations based on the use of linear multistep methods. ACM Trans. Math. Software5, 401–414 (1979) · Zbl 0432.65042 · doi:10.1145/355853.355857
[19] Zlatev, Z., Thomsen, P.G.: Differential integrators based on linear multistep methods. In: Méthodes numérique dans les sciences de l’ingénieur – G.A.M.N.I. (E. Absi, R. Glowinski, P. Lascaux, H. Veysseyre, eds.), Vol. 1, pp. 221–231. Paris: Dunod, 1980
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