Numerical approximation of a wave equation with unilateral constraints

The system u_tt − u_xx ∋ f, x ∈ (0, L) × (0, T), with initial data u(x, 0) = u₀(x), u_t(x, 0) = u₁ (x) almost everywhere on (0, L) and boundary conditions u(0, t) = 0, for all t ≥ 0, and the unilateral condition u_x(L, t) ≥ 0, u(L, t) ≥ k₀, (u(L, t) − k₀) u_x(L, t) = 0 models the longitudinal vibrations of a rod, whose motion is limited by a rigid obstacle at one end. A new variational formulation is given; existence and uniqueness are proved. Finite elements and finite difference schemes are given, and their convergence is proved. Numerical experiments are reported; the characteristic schemes perform better in terms of accuracy, and the subcharacteristic schemes look better.