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  • 1
    Electronic Resource
    Electronic Resource
    Time discretization of parabolic boundary integral equations (1992)
    Springer
    Numerische Mathematik 63 (1992), S. 455-481 
    Details
    ISSN: 0945-3245
    Keywords: 65R20 ; 45A05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In time-dependent boundary integral equations, a boundary element method in space can be coupled with a different type of discretization in time. For the latter a procedure based on linear multistep methods is proposed, which is applicable whenever the Laplace transform of the fundamental solution is known. The stability properties of the method are obtained from those of the underlying multistep method. In the absence of a space discretization, the numerical solution given by the proposed method is identical to that of a semi-discretization in time of the partial differential equation by the underlying multistep method. The theory is presented for the single layer potential equation of the heat equation. Convergence estimates, which are pointwise in time and expressed in terms of the boundary data, are obtained for full discretizations using Galerkin or collocation boundary element methods in space. Numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385870
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  • 2
    Electronic Resource
    Electronic Resource
    Runge-Kutta theory for Volterra integrodifferential equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 119-135 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65R05 ; CR: 5.18
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459081
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations (1994)
    Springer
    Numerische Mathematik 67 (1994), S. 365-389 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M15, 65R20, 65D30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050033
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Rosenbrock methods for differential-algebraic systems with solution-dependent singular matrix multiplying the derivative (1990)
    Springer
    Computing 43 (1990), S. 325-342 
    Details
    ISSN: 1436-5057
    Keywords: 65L05 ; CR: 5.17 ; Rosenbrock methods ; differential-algebraic systems ; singular systems of differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir studieren Konvergenz und Ordnungsbedingungen von Rosenbrock-Verfahren bei deren Anwendung auf differentiell-algebraische Systeme der FormB(y)y′=a(y) mit singulärer MatrixB und konstruieren ein Paar von eingebetteten Verfahren der Ordnungen 3(2).
    Notes: Abstract We study convergence and order conditions of Rosenbrock type methods applied to differential-algebraic systems of the formB(y)y′=a(y), with singular matrixB. An embedded pair of methods of order 3(2) is constructed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02241653
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  • 5
    Electronic Resource
    Electronic Resource
    Integration of stiff mechanical systems by Runge-Kutta methods (1993)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 44 (1993), S. 1022-1053 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter ɛ. Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than ɛ are shown to approximate smooth solutions, and precise error estimates are derived. As ɛ → 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons for this failure are explained, and remedies are proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00942763
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    On the stability of semi-implicit methods for ordinary differential equations (1982)
    Springer
    BIT 22 (1982), S. 211-232 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is to analyze the stability properties of semi-implicit methods (such as Rosenbrock methods,W-methods, and semi-implicit extrapolation methods) for nonlinear stiff systems of differential equations. First it is shown that the numerical solution satisfies ‖y 1‖ ≦ϕ(hμ)‖y 0‖, if the method is applied with stepsizeh to the systemy′ =Ay (μ denotes the logarithmic norm ofA). Properties of the functionϕ(x) are studied. Further, conditions for the parameters of a semi-implicit method are given, which imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh. The restriction on the stepsize, however, does not depend on the stiffness of the differential equation. Finally, the presented theory is applied to the extrapolation method based on the semi-implicit mid-point rule.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01944478
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Multi-grid dynamic iteration for parabolic equations (1987)
    Springer
    BIT 27 (1987), S. 216-234 
    Details
    ISSN: 1572-9125
    Keywords: 65M10 ; 65M15 ; 65M50 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the method which is obtained when a multi-grid method (in space) is first applied directly to a parabolic intitial-boundary value problem, and discretization in time is done only afterwards. This approach is expected to be well-suited to parallel computation. Further, time marching can be done using different time step-sizes in different parts of the spatial domain.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934186
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  • 8
    Electronic Resource
    Electronic Resource
    Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations (1988)
    Springer
    BIT 28 (1988), S. 678-700 
    Details
    ISSN: 1572-9125
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter ε. The coefficients in the expansion of the global error in powers of ε are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01941143
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Error of rosenbrock methods for stiff problems studied via differential algebraic equations (1989)
    Springer
    BIT 29 (1989), S. 77-90 
    Details
    ISSN: 1572-9125
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper studies Rosenbrock methods when they are applied to stiff differential equations containing a small stiffness parameterε. The basic ideas and techniques are the same as those developed for Runge-Kutta methods in an earlier paper of the authors. The results obtained here are essentially those obtained for Diagonally Implicit Runge-Kutta methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932707
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Chebyshev acceleration of picard-lindelöf iteration (1992)
    Springer
    BIT 32 (1992), S. 535-538 
    Details
    ISSN: 1572-9125
    Keywords: 65L05 ; 65D30 ; 65D32
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02074887
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    Paper (German National Licenses)
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