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  • Articles  (34)
  • Other Sources
  • CR: G1.7  (34)
  • 1985-1989  (34)
  • 1930-1934
  • Mathematics  (34)
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  • Articles  (34)
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  • 1
    Electronic Resource
    Electronic Resource
    On implicit Runge-Kutta methods with higher derivatives (1988)
    Springer
    BIT 28 (1988), S. 809-816 
    Details
    ISSN: 1572-9125
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two families of implicit Runge-Kutta methods with higher derivatives are (re-)considered generalizing classical Runge-Kutta methods of Butcher type and f Ehle type. For generalized Butcher methods the characteristic functionG(η) is represented by means of the node polynomial directly, thereby showing that in methods of maximum order,G(η) is connected withs-orthogonal polynomials in exactly the same way as Padé approximations in the classical case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01954901
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  • 2
    Electronic Resource
    Electronic Resource
    Stability analysis of one-step methods for neutral delay-differential equations (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 605-619 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L20 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper stability properties of one-step methods for neutral functional-differential equations are investigate. Stability regions are characterized for Runge-Kutta methods with respect to the linear test equation $$\begin{gathered} y'\left( t \right) = ay\left( t \right) + by\left( {t - \tau } \right) + cy'\left( {t - \tau } \right),t \geqq 0, \hfill \\ y\left( t \right) = g\left( t \right), - \tau \leqq t \leqq 0, \hfill \\ \end{gathered} $$ τ〉0, where,a, b, andc are complex parameters. In particular, it is shown that everyA-stable collocation method for ordinary differential equations can be extended to a method for neutrals delay-differential equations with analogous stability properties (the so called NP-stable method). We also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395814
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  • 3
    Electronic Resource
    Electronic Resource
    Natural Runge-Kutta and projection methods (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 423-438 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65 L 05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently the author defined the class of natural Runge-Kutta methods and observed that it includes all the collocation methods. The present paper is devoted to a complete characterization of this class and it is shown that it coincides with the class of the projection methods in some polynomial spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396327
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  • 4
    Electronic Resource
    Electronic Resource
    Linearly implicit extrapolation methods for differential-algebraic systems (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 197-211 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01406514
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  • 5
    Electronic Resource
    Electronic Resource
    The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 545-563 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398915
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  • 6
    Electronic Resource
    Electronic Resource
    Rosenbrock methods for Differential Algebraic Equations (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 45-63 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper deals with the numerical solution of Differential/Algebraic Equations (DAE) of index one. It begins with the development of a general theory on the Taylor expansion for the exact solutions of these problems, which extends the well-known theory of Butcher for first order ordinary differential equations to DAE's of index one. As an application, we obtain Butcher-type results for Rosenbrock methods applied to DAE's of index one, we characterize numerical methods as applications of certain sets of trees. We derive convergent embedded methods of order 4(3) which require 4 or 5 evaluations of the functions, 1 evaluation of the Jacobian and 1 LU factorization per step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01401021
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  • 7
    Electronic Resource
    Electronic Resource
    On a least-squares collocation method for linear differential-algebraic equations (1988)
    Springer
    Numerische Mathematik 54 (1988), S. 79-90 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L35 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403892
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  • 8
    Electronic Resource
    Electronic Resource
    A note on multistep methods and attracting sets of dynamical systems (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 667-673 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; 34C35 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405195
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  • 9
    Electronic Resource
    Electronic Resource
    An algebraic characterization ofB-convergent Runge-Kutta methods (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 695-705 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405197
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  • 10
    Electronic Resource
    Electronic Resource
    Gleichmäßig einschließende Diskretisierungsverfahren für schwach nichtlineare Randwertaufgaben (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 95-110 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the present paper a new discretization technique for weakly nonlinear boundary value problems, especially for two point boundary value problems, is proposed. By means of the upper and lower discretizations the exact solution can be uniformly bounded from below and from above, respectively.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389432
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