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

In this paper we consider numerical techniques to locate the event points of the differential system x′=f(x)x′=f(x), where ff is a discontinuous vector field along an event surface Σ={x∈Rn|h(x)=0} splitting the state space into two different regions R1R1 and R2R2 and f(x)=fi(x)f(x)=fi(x) when x∈Rix∈Ri, for i=1,2i=1,2 while f1(x)≠f2(x)f1(x)≠f2(x) when x∈Σx∈Σ. Methods based on Adams multistep schemes which approach the event surface ΣΣ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f1f1 (respectively, f2f2) in the region R2R2 (respectively R1R1) and are based on the computation–at each step–of a new time step ττ reducing the value of the event function h(x)h(x) by a fixed quantity.