Numerical approximation of a wave equation with unilateral constraints

Abstract:

The system ${u_{tt}} - {u_{xx}} \backepsilon f$, $x \in (0,L) \times (0,T)$, with initial data $u(x,0) = {u_0}(x)$, ${u_t}(x,0) = {u_1}(x)$ almost everywhere on (0, L) and boundary conditions $u(0,t) = 0$, for all $t \geq 0$, and the unilateral condition \[ {u_x}(L,t) \geq 0,u(L,t) \geq {k_0},(u(L,t) - {k_0}){u_x}(L,t) = 0\] 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.