Runge-Kutta restarters for multistep methods in presence of frequent discontinuities

Differential equations with discontinuities or differential equations coupled to discrete systems require frequent re-initializations of the numerical solution process. The classical starting process of multistep methods, based on increasing the order in the initialization phase, is computationally expensive when frequent discontinuities occur. Instead we propose to use the stage values or weight vectors of these specially constructed explicit Runge-Kutta methods for starting processes. Two practical examples demonstrate these methods.