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Numerical approximation of a wave equation with unilateral constraints

Authors: Michelle Schatzman and Michel Bercovier
Journal: Math. Comp. 53 (1989), 55-79
MSC: Primary 65M25; Secondary 65N30
MathSciNet review: 969491
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Abstract: The system $ {u_{tt}} - {u_{xx}} \mathrel\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

$\displaystyle {u_x}(L,t) \geq 0,u(L,t) \geq {k_0},(u(L,t) - {k_0}){u_x}(L,t) = 0$

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.

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Article copyright: © Copyright 1989 American Mathematical Society