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' = f_1(x) when h(x) 0, and with f1 \neq 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