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  • Artikel  (12)
  • AMS(MOS): 65L05  (12)
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  • Springer  (12)
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  • 1990-1994
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  • 1
    Digitale Medien
    Digitale Medien
    Stability and accuracy of time discretizations for initial value problems (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 245-296 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400542
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  • 2
    Digitale Medien
    Digitale Medien
    Adams methods for neutral functional differential equations (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 221-230 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a “non-smooth” solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01408695
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  • 3
    Digitale Medien
    Digitale Medien
    Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 169-177 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400537
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  • 4
    Digitale Medien
    Digitale Medien
    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 5
    Digitale Medien
    Digitale Medien
    Unconditionally stable explicit methods for parabolic equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 57-68 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396370
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  • 6
    Digitale Medien
    Digitale Medien
    Perturbed collocation and Runge-Kutta methods (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 193-208 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; Cr: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary It is well known thatsome implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order andA, B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural question, ifall RK methods can be considered as a somewhat “perturbed” collocation. After having introduced this notion we give a proof on the order of the method and discuss their stability properties. Much of known theory becomes simple and beautiful.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01397089
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  • 7
    Digitale Medien
    Digitale Medien
    A study of Rosenbrock-type methods of high order (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 279-298 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01397096
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  • 8
    Digitale Medien
    Digitale Medien
    Order conditions for numerical methods for partitioned ordinary differential equations (1981)
    Springer
    Numerische Mathematik 36 (1981), S. 431-445 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR:5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395956
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  • 9
    Digitale Medien
    Digitale Medien
    Stability of explicit time discretizations for solving initial value problems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 61-91 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Stability regions of explicit “linear” time discretization methods for solving initial value problems are treated. If an integration method needsm function evaluations per time step, then we scale the stability region by dividing bym. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for “general” nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396187
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  • 10
    Digitale Medien
    Digitale Medien
    Die Determinantenmethode zur Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 405-420 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.16
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The characteristic exponent ν of the finite Hill differential equation $$y''(x) + \left( {\lambda + \sum\limits_{\kappa = 1}^k {(2t_\kappa ) \cos (2\kappa x)} } \right) y(x) = 0$$ can be evaluated from the relations $$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \frac{{\pi ^2 }}{4} \det C^{(0)} \det S^{(0)}$$ or $$\cos ^2 \left( {\frac{\pi }{2}v} \right) = \det C^{(1)} \det S^{(1)} ,$$ whereS (μ) andC (μ) are certain infinite band matrices. According to Mennicken [3] the convergence of the infinite determinants can be accelerated by splitting up suitable infinite products. In the present paper this method is discussed under numerical aspects, moreover the formulas for the infinite products are simplified in such way that the complex Gamma-function is no longer needed. Finally, the presented determinental method is compared with other methods by means of some numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399008
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  • 11
    Digitale Medien
    Digitale Medien
    Numerical treatment of delay differential equations by Hermite Interpolation (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 235-255 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; 65Q05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically. Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01398255
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  • 12
    Digitale Medien
    Digitale Medien
    High orderP-stable formulae for the numerical integration of periodic initial value problems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 355-370 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy″=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400315
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