Rosenbrock-type Methods Adapted to Differential-algebraic Systems

We consider the numerical solution of differential-algebraic systems of index one given in Kronecker canonical form. The methods described here are derived from the Rosenbrock approach. Hence, they do not require the solution of nonlinear systems of equations but one evaluation of the Jacobian and one LU decomposition per step. By construction, the s-stage method coincides with a solver for nonlinear equations of order i + 1 if the stepsize is set to zero. In this sense, the adaptation to differential-algebraic equations is performed. The special structure of the method leads to simplified order conditions and to an easy implementation. Some particular methods up to order 4 are given. Especially, an embedded 4-stage method of order 4 (3) is derived.