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  • 1
    Electronic Resource
    Electronic Resource
    Algebraic characterization ofI-stable Runge-Kutta methods (1991)
    Springer
    BIT 31 (1991), S. 314-320 
    Details
    ISSN: 1572-9125
    Keywords: 65L20 ; Implicit Runge-Kutta methods ; I-stability ; generalized positive function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Well-known stability concepts for Runge-Kutta methods areA-stability andB-stability. These stability properties can be characterized by algebraic conditions related to the generating matrix of the method. In this note we show, thatI-stable methods can be characterized similarly, yielding aunified description ofB-,A- andI-stability in terms of a matrixR.I-stability, although a weaker concept thanA-stability, is of some relevance in parallelizing Runge-Kutta methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931290
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    An inverse eigenvalue problem: ComputingB-stable Runge-Kutta methods having real poles (1992)
    Springer
    BIT 32 (1992), S. 676-688 
    Details
    ISSN: 1572-9125
    Keywords: Mathematics Subject Classification ; 65F15 ; 65F30 ; 65L20 ; Inverse eigenvalue problem ; eigenvalue assignment ; singly-implicit Runge-Kutta method ; B-stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01994850
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    Paper (German National Licenses)
    Fulltext
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