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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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  • 2
    Electronic Resource
    Electronic Resource
    Efficient sixth order methods for nonlinear oscillation problems (1988)
    Springer
    BIT 28 (1988), S. 898-903 
    Details
    ISSN: 1572-9125
    Keywords: AMS 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of two-step (hybrid) methods is considered for solving pure oscillation second order initial value problems. The nonlinear system, which results on applying methods of this type to a nonlinear differential system, may be solved using a modified Newton iteration scheme. From this class the author has derived methods which are fourth order accurate,P-stable, require only two (new) function evaluations per iteration and have a true real perfect square iteration matrix. Now, we propose an extension to sixth order,P-stable methods which require only three (new) function evaluations per iteration and for which the iteration matrix is a true realperfect cube. This implies that at most one real matrix must be factorised at each step. These methods have been implemented in a new variable step, local error controlling code.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01954909
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  • 3
    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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  • 4
    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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  • 5
    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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