Convergence of continuous approximations for discontinuous ODEs

We investigate a regularisation technique for the solution of discontinuous ordinary differential equations. It is shown that the solutions obtained by the regularisation procedure converge uniformly to an appropriately defined analytical solution whenever the regularisation parameter ϵ goes to zero. Under reasonable conditions the convergence can be shown to be linear in ϵ. Moreover, we present numerical evidence that suggests that the conditions for linear convergence can be further relaxed.