Error of Rosenbrock methods for stiff problems studied via differential algebraic equations (Q1121648)

This paper studies Rosenbrock methods when they are applied to stiff problems in singular perturbation form \(y'=f(y,z),\quad \epsilon z'=g(y,z)\) \((0<\epsilon \ll 1)\) with initial values y(0), z(0) admitting a smooth solution. Let the Rosenbrock methods have classical order p and differential algebraic order r and satisfy a certain condition on the stability function R(\(\omega)\) (e.g. which is fulfilled if the method is A-stable). Then the following interesting convergence results hold: \(y_ n-y(x_ n)=O(h^ r)+O(\epsilon h^ 2),\quad z_ n-z(x_ n)=O(h^ r)+O(\epsilon h),\) and, in the case of \(p>r=2\), \(y_ n-y(x_ n)=O(h^ 3),\quad z_ n-z(x_ n)=O(h^ 2).\) The basic ideas and techniques of the proof are the same as those developed for Runge-Kutta methods in an earlier paper of the authors [ibid. 28, 678-700 (1988; Zbl 0657.65093)], but there is often considerable difference in the details. The results are illustrated by numerical examples.