Numerical event location techniques in discontinuous differential algebraic equations

In this paper, we consider numerical methods for the location of events of differential algebraic equations of index one. These events correspond to cross a discontinuity surface, beyond which another differential algebraic equation holds. Convergence theorems of the numerical event time and event point to the true event time and event point are given. It is proved that, for integrations by semi-implicit methods or Rosenbrock methods, the order of convergence of the numerical event location is the order of convergence of the continuous extension and, for integration by implicit Runge-Kutta methods, the order of convergence is the order of convergence of the implicit RK method.