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  • 1
    Electronic Resource
    Electronic Resource
    Méthodes de Nyström pour l'équation différentielley″=f(x, y) (1976)
    Springer
    Numerische Mathematik 27 (1976), S. 283-300 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé Dans un récent article (Hairer-Wanner [1]) nous avons donné une théorie à l'aide de laquelle on peut facilement calculer les conditions d'ordre pour une méthode de Nyström. Ici nous montrons comment on peut résoudre ce système d'équations non-linéaires. Nous donnons de plus toutes les méthodes d'ordres pours=2, 3, 4 (oùs−1 indique le nombre d'évaluations de la fonction à chaque pas); des méthodes avec un paramètre d'ordres pours=5, 6 et des méthodes particulières d'ordres−1 pours=8, 9.
    Notes: Summary In a recent paper (Hairer-Wanner [1]) we have given a theory with which it is easy to calculate the order conditions for Nyström methods. Here we show how it is possible to solve this system of non-linear algebraic equations. Moreover we present all methods of orders fors=2, 3, 4 (s−1 indicates the number of function evaluations per step); methods with one parameter of orders fors=5, 6 and some special methods of orders−1 fors=8, 9.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396178
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  • 2
    Electronic Resource
    Electronic Resource
    On the order of iterated defect correction (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 409-424 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65LO5 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In a recent article [2] Frank and Überhuber define and motivate the method of iterated defect correction for Runge-Kutta methods. They prove a theorem on the order of that method using the theory of asymptotic expansions. In this paper we give similar results using the theory of Butcher series (see [4]). Our proofs are purely algebraic. We don't restrict our considerations to Runge-Kutta methods, but we admit arbitrary linear one-step methods. At the same time we consider more general defect functions as in [2].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01432878
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  • 3
    Electronic Resource
    Electronic Resource
    Unconditionally stable explicit methods for parabolic equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 57-68 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396370
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  • 4
    Electronic Resource
    Electronic Resource
    Order conditions for numerical methods for partitioned ordinary differential equations (1981)
    Springer
    Numerische Mathematik 36 (1981), S. 431-445 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR:5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of “P-series” is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory″=f(y,y′), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395956
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  • 5
    Electronic Resource
    Electronic Resource
    Dense output for extrapolation methods (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 419-439 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations. In particular, the extrapolated explicit Euler method, the GBS method (for non-stiff equations) and the extrapolated linearly implicit Euler method (for stiff and differential-algebraic equations) are considered. Existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given. Several numerical experiments illustrate the theoretical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385634
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  • 6
    Electronic Resource
    Electronic Resource
    A theory for Nyström methods (1975)
    Springer
    Numerische Mathematik 25 (1975), S. 383-400 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For the numerical solution of differential equations of thesecond order (and systems of ...) there are two possibilities: 1. To transform it into a system of the first order (of doubled dimension) and to integrate by a standard routine. 2. To apply a “direct” method as those invented by Nyström. The benefit of these direct methods is not generally accepted, a historical reason for them was surely the fact that at that time the theories did not consider systems, but single equations only. In any case the second approach is more general, since the class of methods defined in this paper contains the first approach as a special case. So there is more freedom for extending stability or accuracy. This paper begins with the development of a theory, which extends our theory for first order equations [1] to equations of the second order, and which is applicable to the study of possibly all numerical methods for problems of this type. As an application, we obtain Butcher-type results for Nyström-methods, we characterize numerical methods as applications of a certain set of trees, give formulas for a group-structure (expressing the composition of methods) etc. Recently in [2] the equations of conditions for Nyström methods have been tabulated up to order 7 (containing errors). Our approach yields not only the correct equations of conditions in a straight-forward way, but also an insight in the structure of methods that is useful for example in choosing good formulas.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396335
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  • 7
    Electronic Resource
    Electronic Resource
    Unconditionally stable methods for second order differential equations (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 373-379 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 ; CR ; 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2, ...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01401041
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  • 8
    Electronic Resource
    Electronic Resource
    Asymptotic expansions of the global error of fixed-stepsize methods (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 345-360 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65LO5 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In his fundamental paper on general fixed-stepsize methods, Skeel [6] studied convergence properties, but left the existence of asymptotic expansions as an open problem. In this paper we give a complete answer to this question. For the special cases of one-step and linear multistep methods our proof is shorter than the published ones. Asymptotic expansions are the theoretical base for extrapolation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01391413
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  • 9
    Electronic Resource
    Electronic Resource
    A- andB-stability for Runge-Kutta methods-characterizations and equivalence (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 383-389 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 L 20 ; CR: G.1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations ofA-stability andB-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the “exact Runge-Kutta method”. Finally we demonstrate by a numerical example that for difficult problemsB-stable methods are superior to methods which are “only”A-stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389646
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  • 10
    Electronic Resource
    Electronic Resource
    Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 247-258 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 CR ; 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary All rational approximations to exp(z) of order ≧2m−β (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving β real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp≧0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408698
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