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
DOI:
https://doi.org/10.1090/S0025-5718-1989-0969491-5
MathSciNet review:
969491
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Abstract | References | Similar Articles | Additional Information
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.
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Article copyright:
© Copyright 1989
American Mathematical Society