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  • Theta method  (1)
  • absolute stability  (1)
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  • 1
    Digitale Medien
    Digitale Medien
    A-Stability and Stochastic Mean-Square Stability (2000)
    Springer
    BIT 40 (2000), S. 404-409 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Mean-square ; Milstein method ; multiplicative noise ; Theta method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability property of the Theta Method does not carry through to the mean-square context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A(α)-stability that may be useful in the stability analysis of more general methods. It is also shown that mean-square A-stability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1022355410570
    Bibliothek Standort Signatur Band/Heft/Jahr Verfügbarkeit
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  • 2
    Digitale Medien
    Digitale Medien
    Runge-Kutta stability on a Floquet problem (1994)
    Springer
    BIT 34 (1994), S. 88-98 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Runge-Kutta ; absolute stability ; Floquet ; equilibrium ; steady state ; 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This work examines the stability of explicit Runge-Kutta methods applied to a certain linear ordinary differential equation with periodic coefficients. On this problem naïve use of the eigenvalues of the Jacobian results in misleading conclusions about stable behaviour. It is shown, however, that a valid analogue of the classical absolute stability theory can be developed. Further, using a suitable generalisation of the equilibrium theory of Hall [ACM Trans. on Math. Soft. 11 (1985), pp. 289–301], accurate predictions are made about the performance of modern, adaptive algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01935018
    Bibliothek Standort Signatur Band/Heft/Jahr Verfügbarkeit
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