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  • Articles  (123)
  • Other Sources
  • CR: 5.17  (97)
  • AMS(MOS): 65L05  (41)
  • 2010-2014
  • 1985-1989  (21)
  • 1980-1984  (102)
  • Mathematics  (123)
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  • Articles  (123)
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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
    Superstable implicit Runge-Kutta methods for second order initial value problems (1989)
    Springer
    BIT 29 (1989), S. 518-526 
    Details
    ISSN: 1572-9125
    Keywords: AMS(MOS): 65L05 ; Second order initial value problems ; region of absolute stability ; interval of periodicity ; interval of (weak) stability ; superstable methods ; implicit Runge-Kutta methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy″=f(x, y, y′),y(x o)=y o,y′(x o)=y′ o. These methods when applied to the test equationy″+2αy′+β 2 y=0, α,β≥0, α+β〉0, are superstable with the exception of a finite number of isolated values ofβh. These methods can be successfully used for solving singular perturbation problems for which δf/δy and/or δf/δy′ are negative and large. Numerical results demonstrate the efficiency of these methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02219237
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  • 3
    Electronic Resource
    Electronic Resource
    Extrapolation at stiff differential equations (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 377-400 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G.1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied tostiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour ofextrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01462235
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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
    Contractivity preserving explicit linear multistep methods (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 213-223 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; 65L20 ; 65M10 ; CR: G.1.7, G.1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on the general test-equation $$\frac{d}{{dt}}U(t) = AU(t)$$ , whereA is a square matrix of arbitrary orders≧1. The contractivity is analysed with respect to arbitrary norms in thes-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01406515
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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
    Approximation of time-dependent free boundaries (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 179-199 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; Primary 65N30 ; Secondary 35R35 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Two problems are considered in the paper: the first of them is connected with elliptic variational inequalities and consists in developing a moving obstacle algorithm for approximating the unknown free boundary; the other problem is linked with numerical solution of the Stefan problem, which is formulated in the similar way as in the elliptic case. Some computational aspects are also discussed in the paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400538
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