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 x ′ = f 2 ( x ) when h ( x ) > 0 , and with f 1 ≠ f 2 for x ∈ Σ , where Σ : = { x : h ( x ) = 0 } is a smooth co-dimension one discontinuity surface. Often, f 1 and f 2 are defined on the whole space, but there are applications where f 1 is not defined above Σ and f 2 is not defined below Σ. For this reason, we consider explicit Runge–Kutta methods which do not evaluate f 1 above Σ (respectively, f 2 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.