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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
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