Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side
This paper deals with the numerical solution of initial value problems (IVPs) (1) \( x'(t) = f(x(t)), ~x(0)=x_0 \in {\mathbb R}^d\) where \(f\) is a discontinuous vector field defined by \( f(x)= f_1(x)\) if \( x \in S_1 = \{ x ; h(x)>0\} \) and \( f(x)=f_2(x)\) if \( x \in S_2 = \{ x ; h(x)< 0\} \) with a known discontinuity hypersurface in the \(d\)-dimensional space defined by \( \{x \in R^d; h(x)=0 \}\) where \( f_1\) may not be defined in \(S_2\) and \(f_2\) in \(S_1\). After some comments on the extension of these discontinuous systems into Filippov convex differential inclusions that allows one to define solutions that may slide along the discontinuity hypersurface or cross it, the authors consider the so-called one-sided explicit Runge-Kutta methods for solving one-sided solutions of IVPs (1). Such methods permit to define numerically a solution of (1) up to the point that hits the discontinuous hypersurface and are defined by adjusting the stepsize so that the vector field is computed only in the same side of the hypersurface. Some particular Runge-Kutta methods up to fourth order are considered and the numerical experiments with two elementary test problems are presented to show the performance of the methods.
scientific article; zbMATH DE number 6155392