Event location for ordinary differential equations

An initial value problem for y′ = f(t, y) may have an associated event function g(t, y). An event is said to occur at t* when g(t*, y(t*)) = 0. We consider problems for which the definition of f(t, y) changes at the time of an event. A number of solvers locate events and restart the integration there so as to deal with the changes in f, but there is little theoretical support for what is done. Here we prove that with reasonable assumptions about the problem and the solver, the error of the numerical solution is qualitatively the same whether or not events occur. Numerical results obtained with a wide range of solvers confirm the theory developed here.