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  • Artikel: DFG Deutsche Nationallizenzen  (15)
  • 1
    Digitale Medien
    Digitale Medien
    Stability analysis of one-step methods for neutral delay-differential equations (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 605-619 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L20 ; CR: G1.7
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395814
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  • 2
    Digitale Medien
    Digitale Medien
    Stability analysis of product θ-methods for Abel integral equations of the second kind (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 127-136 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65R20 ; CR: G1.9
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Stability analysis of product θ-methods for Abel integral equations of the second kind, based on the test equation $$y(t) = 1 - \lambda \int\limits_0^1 {\frac{{y(s)}}{{(t - s)^\alpha }}} ds,t \geqq 0,$$ 0〈α〈1, is presented. It is known that the solutionY of this equation is bounded if and only if λ〉0 and we investigate whether this property is inherited by numerical approximations toY.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389867
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  • 3
    Digitale Medien
    Digitale Medien
    Stability of reducible quadrature methods for Volterra integral equations of the second kind (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 159-173 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65R20 ; CR: G1.9
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Stability analysis of reducible quadrature methods for Volterra integral equations based on the test equation $$y(t) = 1 + \int\limits_0^t {(\lambda + \mu t + vs)y(s)ds,t \geqq 0} $$ is presented. The concept of absolute stability is defined and necessary and sufficient conditions for the method to be absolutely stable for given λ, μ, andv are derived. These conditions are illustrated for the class of θ-methods for integral equations. The main tool in our stability analysis is the theory of difference equations of Poincaré type.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389707
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  • 4
    Digitale Medien
    Digitale Medien
    On the convergence of multistep methods for the Cauchy problem for ordinary differential equations (1978)
    Springer
    Computing 20 (1978), S. 351-361 
    Details
    ISSN: 1436-5057
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik
    Beschreibung / Inhaltsverzeichnis: Zusammenfassung Es wird ein allgemeines, quasilineares, nichtstationäresk-Schrittverfahren für die Lösung des Cauchy-Problems für gewöhnliche Differentialgleichungen untersucht. Ein Konvergenzsatz mit ziemlich schwachen Bedingungen wird angegeben. Die Inkrementfunktion muß nicht Lipschitz-stetig sein; es genügt, wenn diese Funktion die Perron-Bedingung aus der Eindeutigkeitstheorie für das Cauchy-Problem mit nichtabnehmender Vergleichfunktion erfüllt. Das Ergebnis ist eine Erweiterung der Theorie von G. Dahlquist und des letzten Resultats von K. Taubert.
    Notizen: Abstract The general form of a quasilinear nonstationaryk-step method for solving of the Cauchy problem for ordinary differential equations is discussed. The convergence theorem is stated under rather weak conditions. It is not assumed that the increment function is Lipschitz-continuous but only that it satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function. The result established in the paper is an extension of the theory given by G. Dahlquist and the recent result of K. Taubert.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02252383
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  • 5
    Digitale Medien
    Digitale Medien
    Global stability analysis of the Runge-Kutta methods for volterra integral and integro-differential equations with degenerate kernels (1990)
    Springer
    Computing 45 (1990), S. 291-300 
    Details
    ISSN: 1436-5057
    Schlagwort(e): Primary 65R20 ; Secondary 45D05 ; 45L10 ; Volterra integral equation ; Volterra integro-differential equation ; Degenerate kernel ; Stability ; Runge-Kutta method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik
    Beschreibung / Inhaltsverzeichnis: Zusammenfassung Wir untersuchen die Stabilität der numerischen Lösung von Volterraschen Integral- und Integral-Differentialgleichungen mit degeneriertem Kern mit Hilfe von ganz allgemeinen Klassen von Runge-Kutta Methoden. Die Resultate sind Verallgemeinerungen früherer Resultate, die von den Autoren für die exakte Kollokationsmethode für diese Gleichungen erhalten worden sind.
    Notizen: Abstract We investigate the stability of the numerical solutions resulting from applying very general classes of Runge-Kutta methods to Volterra integral and integro-differential equations with degenerate kernels. The results are generalizations of previous results obtained by the authors for exact collocation methods for these equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02238797
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  • 6
    Digitale Medien
    Digitale Medien
    Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods (1992)
    Springer
    BIT 32 (1992), S. 104-117 
    Details
    ISSN: 1572-9125
    Schlagwort(e): AMS 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In this paper it is shown that the local discretization error ofs-stage singly-implicit methods of orderp can be estimated by embedding these methods intos-stage two-step Runge-Kutta methods of orderp+1, wherep=s orp=s+1. These error estimates do not require any extra evaluations of the right hand side of the differential equations. This is in contrast with the error estimation schemes based on embedded pairs of two singly-implicit methods proposed by Burrage.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01995111
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  • 7
    Digitale Medien
    Digitale Medien
    General linear methods with external stages of different orders (1996)
    Springer
    BIT 36 (1996), S. 688-712 
    Details
    ISSN: 1572-9125
    Schlagwort(e): General linear methods ; order conditions ; stage errors ; error estimation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01733788
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  • 8
    Digitale Medien
    Digitale Medien
    Diagonally implicit general linear methods for ordinary differential equations (1993)
    Springer
    BIT 33 (1993), S. 452-472 
    Details
    ISSN: 1572-9125
    Schlagwort(e): AMS(MOS) ; 65L05 ; 65L07 ; General linear method ; order conditions ; stability analysis
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Padé approximations to the exponential function.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01990528
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  • 9
    Digitale Medien
    Digitale Medien
    Preconditioning waveform relaxation iterations for differential systems (1996)
    Springer
    BIT 36 (1996), S. 54-76 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Waveform relaxation ; splittings ; preconditioning ; overlapping ; error analysis ; parallel computing
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We discuss preconditioning and overlapping of waveform relaxation methods for sparse linear differential systems. It is demonstrated that these techniques significantly improve the speed of convergence of the waveform relaxation iterations resulting from application of various modes of block Gauss-Jacobi and block Gauss-Seidel methods to differential systems. Numerical results are presented for linear systems resulting from semi-discretization of the heat equation in one and two space variables. It turns out that overlapping is very effective for the system corresponding to the one-dimensional heat equation and preconditioning is very effective for the system corresponding to the two-dimensional case.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01740544
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  • 10
    Digitale Medien
    Digitale Medien
    Construction of Runge–Kutta methods of Crouch–Grossman type of high order (2000)
    Springer
    Advances in computational mathematics 13 (2000), S. 405-415 
    Details
    ISSN: 1572-9044
    Schlagwort(e): Runge–Kutta methods ; order conditions ; geometric integration ; rigid frames ; least squares minimization
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1016645730465
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