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  • 1
    Electronic Resource
    Electronic Resource
    Order stars and stability for delay differential equations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 371-383 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We consider Runge–Kutta methods applied to delay differential equations $y'(t)=ay(t)+by(t-1)$ with real a and b. If the numerical solution tends to zero whenever the exact solution does, the method is called $\tau (0)$ -stable. Using the theory of order stars we characterize high-order symmetric methods with this property. In particular, we prove that all Gauss methods are $\tau (0)$ -stable. Furthermore, we present sufficient conditions and we give evidence that also the Radau methods are $\tau (0)$ -stable. We conclude this article with some comments on the case where a andb are complex numbers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050454
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Backward error analysis for multistep methods (1999)
    Springer
    Numerische Mathematik 84 (1999), S. 199-232 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L06
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In recent years, much insight into the numerical solution of ordinary differential equations by one-step methods has been obtained with a backward error analysis. It allows one to explain interesting phenomena such as the almost conservation of energy, the linear error growth in Hamiltonian systems, and the existence of periodic solutions and invariant tori. In the present article, the formal backward error analysis as well as rigorous, exponentially small error estimates are extended to multistep methods. A further extension to partitioned multistep methods is outlined, and numerical illustrations of the theoretical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050469
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A necessary condition forBSI-stability (1985)
    Springer
    BIT 25 (1985), S. 285-288 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper gives a negative result onBSI-stability for the Lobatto IIIC-methods with more than two stages. We also give a necessary condition forBSI-stability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01935006
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A note onD-stability (1984)
    Springer
    BIT 24 (1984), S. 383-386 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper gives new insight into the concept ofD-stability of Runge-Kutta methods for stiff ordinary differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02136038
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Asymptotic Error Analysis of the Adaptive Verlet Method (1999)
    Springer
    BIT 39 (1999), S. 25-33 
    Details
    ISSN: 1572-9125
    Keywords: Adaptive Verlet method ; time-reversible variable stepsizes ; Hamiltonian systems ; Sundman time-transformations ; backward error analysis ; asymptotic expansions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Adaptive Verlet method and variants are time-reversible schemes for treating Hamiltonian systems subject to a Sundman time transformation. These methods have been observed in computer experiments to exhibit superior numerical stability when implemented in a counterintuitive “reciprocal” formulation. Here we give a theoretical explanation of this behavior by examining the leading terms of the modified equation (backward error analysis) and those of the asymptotic error expansion. With this insight we are able to improve the algorithm by simply correcting the starting stepsize.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022313123291
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    Paper (German National Licenses)
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