Stabilized starting algorithms for collocation Runge-Kutta methods

In this paper, we propose a technique to stabilize some starting algorithms often used in the Newton-type iterations appearing when collocation Runge-Kutta methods are applied to solve stiff initial value problems. By following the ideas given in [1], we analyze the order (classical and stiff) of the new starting algorithms and pay special attention to their error amplifying functions. From the computational point of view, the new algorithms require the solution of an additional linear system per integration step, but as shown in the numerical experiments, this extra cost is compensated in most of the problems by their better stability properties.