About one new two-stages Rosenbrock scheme for differential-algebraic systems

The equations for coefficients of two-stages Rosenbrock scheme guarantying approximation with third order of accuracy for differential-algebraic system (index 1) are obtained in this paper. These coefficients are complex numbers. New 2-stages Rosenbrock scheme is constructed. It is L2-stable. This scheme has accuracy $O(\tau^3)$ for differential-algebraic systems and $O(\tau^4)$ for pure differential stiff systems. Convergence of this scheme is proved. This scheme was tested on several standard stiff tests and compared with previously know scheme of the same class.