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**Second derivative methods with RK stability.**
*(English)*
Zbl 1084.65069

Authors’ summary: General linear methods are extended to the case in which second derivatives, as well as first derivatives, can be calculated. Methods are constructed of third and fourth order which are A-stable, possess the Runge-Kutta (RK) stability property and have a diagonally implicit structure for efficient implementation.

Reviewer’s remark: Such methods are at an early stage of devlopment but the present results, and the fact that there is a fair amount of freedom in choosing method parameters, show these methods deserve further development. The issues of starting procedure, error estimation, and stepsize control remain to be addressed.

Reviewer’s remark: Such methods are at an early stage of devlopment but the present results, and the fact that there is a fair amount of freedom in choosing method parameters, show these methods deserve further development. The issues of starting procedure, error estimation, and stepsize control remain to be addressed.

Reviewer: John Pryce (Swindon)

### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

### Keywords:

stiff ODEs; multistep and multiderivative methods; general linear methods; A-stability; L-stability; RK-stability; Runge-Kutta stability
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\textit{J. C. Butcher} and \textit{G. Hojjati}, Numer. Algorithms 40, No. 4, 415--429 (2005; Zbl 1084.65069)

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### References:

[1] | J.C. Butcher, General linear methods for stiff differential equations, BIT 41 (2001), 240–264. · Zbl 0983.65085 |

[2] | J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, New York, 2003). · Zbl 1040.65057 |

[3] | J.C. Butcher and W.M. Wright, The construction of practical general linear methods, BIT 43 (2003) 695–721. · Zbl 1046.65054 |

[4] | J.R. Cash, Second derivative extended backward differentiation formulas for the numerical integration of stiff systems, SIAM J. Numer. Anal. 18(2) (1981) 21–36. · Zbl 0452.65047 |

[5] | G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963) 27–43. · Zbl 0123.11703 |

[6] | G. Hojjati, M.Y. Rahimi Ardabili and S.M. Hosseini, New second derivative multistep methods for stiff systems, Appl. Math. Modelling (to appear). · Zbl 1101.65078 |

[7] | D. Lee and S. Preiser, A class of nonlinear multistep numerical A-stable methods for solving stiff differential equations, Comput. Math. Appl. 4 (1978) 43–51. · Zbl 0377.65028 |

[8] | W.M. Wright, General linear methods with inherent Runge–Kutta stability, Ph.D. thesis, The University of Auckland (2003). · Zbl 1016.65049 |

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