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  • 1
    Electronic Resource
    Electronic Resource
    A High Accuracy Defect-Correction Multigrid Method for the Steady Incompressible Navier-Stokes Equations
    Amsterdam : Elsevier
    Journal of Computational Physics 114 (1994), S. 227-233 
    Details
    ISSN: 0021-9991
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Computer Science , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/jcph.1994.1162
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  • 2
    Electronic Resource
    Electronic Resource
    A study of B-convergence of Runge-Kutta methods (1986)
    Springer
    Computing 36 (1986), S. 17-34 
    Details
    ISSN: 1436-5057
    Keywords: 65L05 ; 5.17 ; Numerical analysis ; implicit Runge-Kutta methods ; stiff problems ; B-convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Dieser Aufsatz befaßt sich mit der Analyse der Konvergenz von impliziten Runge-Kutta Verfahren für steife, semi-lineare Systeme der Form $$\dot U$$ (t)=QU(t)+g(t, U(t)). Ein Kriterium wird entwickelt, welches entscheidet, ob die Ordnung der optimalenB-Konvergenz mindestens gleich der Stufenordnung oder um eine Ordnung höher ist. Dieses Kriterium wird untersucht für eine Zahl von interessanten Klassen von Verfahren.
    Notes: Abstract This paper deals with the convergence analysis of implicit Runge-Kutta methods as applied to stiff, semilinear systems of the form $$\dot U$$ (t)=QU(t)+g(t, U(t)). A criterion is developed which determines whether the order of optimalB-convergence is at least equal to the stage order or one order higher. This criterion is studied for a number of interesting classes of methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02238189
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  • 3
    Electronic Resource
    Electronic Resource
    Preconditioning waveform relaxation iterations for differential systems (1996)
    Springer
    BIT 36 (1996), S. 54-76 
    Details
    ISSN: 1572-9125
    Keywords: Waveform relaxation ; splittings ; preconditioning ; overlapping ; error analysis ; parallel computing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01740544
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  • 4
    Electronic Resource
    Electronic Resource
    The order ofB-convergence of algebraically stable Runge-Kutta methods (1987)
    Springer
    BIT 27 (1987), S. 62-71 
    Details
    ISSN: 1572-9125
    Keywords: 65L05 ; Numerical analysis ; stiff initial value problems ; implicit Runge-Kutta methods ; B-convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a previous paper it was shown that for a class of semi-linear problems many high order Runge-Kutta methods have order of optimalB-convergence one higher than the stage order. In this paper we show that for the more general class of nonlinear dissipative problems such as result holds only for a small class of Runge-Kutta methods and that such methods have at most classical order 3.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01937355
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  • 5
    Electronic Resource
    Electronic Resource
    Optimizing simulation models of agricultural systems (1998)
    Springer
    Annals of operations research 82 (1998), S. 219-232 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Agricultural systems vary widely in terms of scale, scope and purpose. Managers of thesereal-world systems are typically faced with a multitude of alternative management optionsand strategies, and are turning more towards simulation models in an attempt to evaluatethese and identify the optimal combination. From a modelling perspective, agriculturalsystems present a range of problems which need to be addressed, and these are outlinedwith examples. General conclusions are drawn on which of the available methodologies aremost likely to be successful for users.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018958602679
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  • 6
    Electronic Resource
    Electronic Resource
    Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations (2000)
    Springer
    BIT 40 (2000), S. 451-470 
    Details
    ISSN: 1572-9125
    Keywords: Stochastic ODEs ; strong convergence ; linear multistep formulae ; Adams methods ; predictor-corrector methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The modelling of many real life phenomena for which either the parameter estimation is difficult, or which are subject to random noisy perturbations, is often carried out by using stochastic ordinary differential equations (SODEs). For this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of linear multistep formulae (LMF). Strong order convergence conditions up to order 1 are stated, for both commutative and non-commutative problems. The case of additive noise is further investigated, in order to obtain order improvements. The implementation of the methods is also considered, leading to a predictor-corrector approach. Some numerical tests on problems taken from the literature are also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022363612387
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  • 7
    Electronic Resource
    Electronic Resource
    Stochastic Methods for Ill-Posed Problems (2000)
    Springer
    BIT 40 (2000), S. 226-240 
    Details
    ISSN: 1572-9125
    Keywords: Stochastic differential equations ; regularisation ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one-step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as α = $$\sqrt {\tau _n }$$ , where τ n is the stepsize and provides the convergence on smooth initial data. The case of the approximation of well-posed problems is also considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022386822865
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  • 8
    Electronic Resource
    Electronic Resource
    Parallel iterated methods based on variable step‐size multistep Runge—Kutta methods of Radau type for stiff problems (2000)
    Springer
    Advances in computational mathematics 13 (2000), S. 257-270 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In our previous paper [3], the performance of a variable step‐size implementation of Parallel Iterated Methods based on Multistep Runge–Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep Runge–Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step‐size basis. Similar unsatisfactory results based on this method were also noted by Schneider [12], who showed that the method is only suitable when the step‐size does not vary too often. In this paper, we design the Variable step‐size Multistep Runge–Kutta (VMRK) method as the underlying formula for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step‐size MRK (PIVMRK) methods improve substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge–Kutta methods (PIRKs).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018906311204
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  • 9
    Electronic Resource
    Electronic Resource
    Parallel iterated methods based on multistep Runge-Kutta methods of Radau type (1997)
    Springer
    Advances in computational mathematics 7 (1997), S. 37-57 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper investigates iterated Multistep Runge-Kutta methods of Radau type as a class of explicit methods suitable for parallel implementation. Using the idea of van der Houwen and Sommeijer [18], the method is designed in such a way that the right-hand side evaluations can be computed in parallel. We use stepsize control and variable order based on iterated approximation of the solution. A code is developed and its performance is compared with codes based on iterated Runge-Kutta methods of Gauss type and various Dormand and Prince pairs [15]. The accuracy of some of our methods are comparable with the PIRK10 methods of van der Houwen and Sommeijer [18], but require fewer processors. In addition at very stringent tolerances these new methods are competitive with RK78 pairs in a sequential implementation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018930415863
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  • 10
    Electronic Resource
    Electronic Resource
    Parallel iterated method based on multistep Runge-Kutta of Radau type for stiff problems (1997)
    Springer
    Advances in computational mathematics 7 (1997), S. 59-77 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Research on parallel iterated methods based on Runge-Kutta formulas both for stiff and non-stiff problems has been pioneered by van der Houwen et al., for example see [8-11]. Burrage and Suhartanto have adopted their ideas and generalized their work to methods based on Multistep Runge-Kutta of Radau type [2] for non-stiff problems. In this paper we discuss our methods for stiff problems and study their performance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018982432701
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