IRK methods for DAE: starting algorithms

When semi-explicit differential-algebraic equations are solved with implicit Runge–Kutta methods, the computational effort is dominated by the cost of solving the nonlinear systems. That is why it is important to have good starting values to begin the iterations. In this paper we study a type of starting algorithms, without additional computational cost, in the case of index-1 DAE. The order of the starting values is defined, and by using DA-series and rooted trees we obtain their general order conditions. If the RK satisfies some simplified assumptions, then the maximum order can be obtained.