A vibrating string with dynamic frictionless impact

Abstract

In this paper, we consider the wave equation with unilateral constraints, which describes the transverse motion of vibrating string with fixed ends impacting a stationary rigid obstacle. Conservation of energy is investigated. Convergence of the numerical trajectories in both semi-discrete case and fully-discrete case is proved. The numerical scheme is implemented with the complementarity problem that arises in the numerical method being solved, using the Fischer–Burmeister function. Our numerical results provide evidence of energy conservation with some particular shapes of obstacle.