The numerical solution of linear time-varying DAEs with index 2 by IRK methods

Differential-algebraic equations (DAEs) with a higher index can be approximated by implicit Runge-Kutta methods (IRK). Until now,.a number of initial value problems have been approximated by Runge-Kutta methods, but all these problems have a special semi-explicit or Hessenberg form. In the present paper we consider IRK methods applied to general linear time-varying (nonautonomous) DAEs tractable with index 2. For some stiffly accurate IRK formulas we show that the order of accuracy in the differential component is the same nonstiff order, if the DAE has constant nullspace. We prove that IRK methods cannot be feasible or become exponentially unstable when applied to linear DAEs with variable nullspace. In order to overcome these difficulties we propose a new approach for this case. Feasibility, weak instability and convergence are proved. Order results are given in terms of the Butcher identities.