A numerical scheme for impact problems I: The one-dimensional case

We consider a mechanical system with impact and one degree of freedom. The system is not necessarily Lagrangian. The representative point is subject to the constraint u(t) is an element of R+ for all t. We assume that, at impact, the velocity is reversed and multiplied by a given coefficient of restitution e [ 0, 1]. We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution. Many of the features of this proof will be reused in the nonconvex, multidimensional case, written in generalized coordinates, given in the companion paper [L. Paoli and M. Schatzman, SIAM J. Numer. Anal., 40 ( 2002), pp. 734 768]. We present some numerical results obtained with the scheme for a spring-dashpot system and we compare them to the results obtained by impact detection and penalization.