A Numerical Scheme for Impact Problems

We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point is subject to a constraint: it must stay inside a closed set K with boundary of class C. We assume that, at impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution e ∈ [0, 1]: the mechanically relevant notion of orthogonality is defined in terms of the local metric for the impulsions (local cotangent metric). 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, which yields also an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. The technique of proof uses a localization of the scheme close to the boundary of K; this idea is classical for a differential system studied in the framework of flows of a vector field; it is much more difficult to implement here, because finite differences schemes are only approximately local: straightening the boundary creates quadratic terms which cause all the difficulties of the proof.