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**Stability of the Rosenbrock methods for the neutral delay differential-algebraic equations.**
*(English)*
Zbl 1081.65079

The authors derive the Rosenbrock methods for a class of neutral delay differential-algebraic equations (NDDAEs) from natural semi-implicit Runge-Kutta methods for general NDDAEs. Under certain stated assumptions they show that the method is GP-stable. A numerical example illustrating the GP-stability of a Rosenbrock method is included. The authors include a brief introduction to Rosenbrock methods, including relevant definitions relating to stability. References pertaining to the stability of numerical methods are given, with a particular emphasis on the Rosenbrock methods. Results relating to the solvability of linear constant coefficient DAEs and to the asymptotic stability of their solution are stated and a theorem giving sufficient conditions for the Rosenbrock method to be asymptotically stable if the linear constant coefficient DAE is asymptotically stable is given. The GP-stability of Runge-Kutta methods for NDDAEs is also considered.

Reviewer: Pat Lumb (Chester)

### MSC:

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

34K40 | Neutral functional-differential equations |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

65L80 | Numerical methods for differential-algebraic equations |

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

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

### Keywords:

Rosenbrock methods; stability; neutral delay differential algebraic equations; Runge-Kutta method; numerical example
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\textit{J. J. Zhao} et al., Appl. Math. Comput. 168, No. 2, 1128--1144 (2005; Zbl 1081.65079)

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### References:

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