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  • 1
    Electronic Resource
    Electronic Resource
    Efficient fourth orderP-stable formulae (1987)
    Springer
    BIT 27 (1987), S. 599-614 
    Details
    ISSN: 1572-9125
    Keywords: MAS ; 65L05 ; Second order initial value problems ; oscillation problems ; efficientP-stable methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01937279
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Damping and phase analysis for some methods for solving second-order ordinary differential equations (1983)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 19 (1983), S. 495-503 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider numerical methods for initial value problems for second-order systems of ordinary differential equations, analysing them by applying them to the test equation We discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property. We also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation.To illustrate the theory, we consider the damping and phase properties of some frequently used methods.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620190404
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The methods of Gellert and of Brusa and Nigro are Padé approximant methods (1984)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 20 (1984), S. 1307-1322 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We discuss properties of methods proposed recently by Gellert1 and by Brusa and Nigro2 for solving systems of second-order ordinary differential equations. In both cases, we extend the derivations to obtain whole families of methods with improved stability properties (roughly equivalent to L-stability for first-order systems) and we show that obtaining improvement in stability properties results in a deterioration in accuracy and/or implementation properties. Also we bring out the relationship (and in some cases equivalence) of these methods to others proposed in the literature. In particular, we show that the methods proposed by the above authors are essentially equivalent to well-known methods of long standing, namely Padé approximant methods for equivalent first-order systems of ordinary differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620200710
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  • 4
    Electronic Resource
    Electronic Resource
    Variable-order variable-step algorithms for second-order systems. Part 2: The codes (1988)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 26 (1988), S. 55-80 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In a companion paper, Thomas and Gladwell,4 the authors designed a class of multistage methods for second-order systems of ordinary differential equations (ODE's), together with local error estimators based on embedding techniques. Here we provide a set of subroutines implementing the algorithms and discuss our numerical experience with the resulting codes.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620260105
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  • 5
    Electronic Resource
    Electronic Resource
    Variable-order variable-step algorithms for second-order systems. Part 1: The methods (1988)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 26 (1988), S. 39-53 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We develop a new class of multistage algorithms for second-order systems of ordinary differential equations (ODE's) based on methods by Zienkiewicz et al.13 Our new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques. In a companion paper, we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes.
    Additional Material: 7 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620260104
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  • 6
    Electronic Resource
    Electronic Resource
    A family of Houbolt methods for dynamic vibration problems (1983)
    New York, NY [u.a.] : Wiley-Blackwell
    International Journal for Numerical and Analytical Methods in Geomechanics 7 (1983), S. 321-336 
    Details
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: We extend the well-known Houblt method for dynamic vibration problems to a class of methods suitable for a variable-order variable-step computer program implementation. We discuss the numerical stability properties of these methods and derive a family of extensions with improved stability properties. Finally we present some numerical results which compare a variety of Houbolt methods with other well-known Backward Differentiation methods, demonstrating in passing the practical value of the stability, concepts.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nag.1610070305
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    A single-step algorithm for oscillatory problems (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Communications in Applied Numerical Methods 5 (1989), S. 113-120 
    Details
    ISSN: 0748-8025
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider a class of multistage single-step methods introduced by Zienkiewicz et al.1 and developed by Thomas and Gladwell.2 From this class, we derive methods which are particularly suitable for the numerical solution of oscillatory problems. We propose a particular pair of methods which permit local error estimation while retaining a ‘one-step’ mode of operation. We discuss the implementation of this pair for both linear and nonlinear problems, and illustrate the discussion with numerical results.
    Additional Material: 5 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1630050208
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    Paper (German National Licenses)
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