Time-transformations for the event location in discontinuous ODEs

In this paper, we consider numerical methods for the location of events of ordinary differential equations. These methods are based on particular changes of the independent variable, called time-transformations. Such a time-transformation reduces the integration of an equation up to the unknown point, where an event occurs, to the integration of another equation up to a known point. This known point corresponds to the unknown point by means of the time-transformation. This approach extends the one proposed by Dieci and Lopez [BIT 55 (2015), no. 4, 987–1003], but our generalization permits, amongst other things, to deal with situations where the solution approaches the event in a tangential way. Moreover, we also propose to use this approach in a different manner with respect to that of Dieci and Lopez.