Direct event location techniques based on Adams multistep methods for discontinuous ODEs

We investigate numerical techniques to locate the event points of the differential system x′ = f(x), characterized by a vector field f which is a discontinuous function along an event surface Σ = {x ∈ R| h(x) = 0} splitting the state space into two different regions R1 and R2, particularly f(x) = fi(x) when x ∈ Ri, for i = 1, 2 and f1(x) 6= f2(x) when x ∈ Σ. We propose event location techniques which approach the event surface Σ from one side only and in a finite number of steps and which use numerical trajectory obtained by Adams multistep methods. The numerical methods proposed do not require the evaluation of the vector field f1 (respectively f2) in the region R2 (respectively R1) and is mainly based on the idea to compute –at each step– the new time step τ of the method so that the event function h(x) is reduced by a fixed quantity.