Pulse detection software for initial value ODEs

In many physical models ordinary differential equations (ODEs) arise with the general form, y′(t) = f(t,y) + g(t), in which abrupt but large changes of limited duration, known as pulses, occur in g(t). These pulses may begin at times which are not known beforehand and may have unknown durations. If the duration is sufficiently short, standard differential equation solvers may miss the pulse completely, stepping over it, especially if, prior to the pulse, the solution is well behaved. In this paper, we discuss software which employs standard initial value ODE software and a process of detect sampling to attempt to detect, and handle efficiently, any pulses which arise. A key advantage of this software and the algorithms for pulse detection and handling described in this paper is that they do not involve modification of the initial value ODE solver. The performance of the new software will be investigated by applying it to several test problems exhibiting pulses. The results show that pulses can be detected and efficiently handled by the new software and that significant computational savings are achieved.