Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side

We consider the numerical integration of discontinuous differential systems of ODEs of the type: x′=f1(x) when h(x)<0 and x′=f2(x) when h(x)>0, and with f1≠f2 for x∈Σ, where Σ:={x:h(x)=0} is a smooth co-dimension one discontinuity surface. Often, f1 and f2 are defined on the whole space, but there are applications where f1 is not defined above Σ and f2 is not defined below Σ. For this reason, we consider explicit Runge–Kutta methods which do not evaluate f1 above Σ (respectively, f2 below Σ). We exemplify our approach with subdiagonal explicit Runge–Kutta methods of order up to 4. We restrict attention only to integration up to the point where a trajectory reaches Σ.