Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] L. Amerio, "Su un problema di vincoli unilaterali per l'equazione non omogenea della corda vibrante," I.A.C. (Istituto per le Applicazioni del Calcolo "Mauro Picone") Pubbl. Ser. D., v. 109, 1976, pp. 3-11.
  • [2] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • [3] Claudio Citrini, The energy theorem in the impact of a string vibrating against a pointshaped obstacle, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 62 (1977), no. 2, 143–149 (English, with Italian summary). MR 0499746
  • [4] C. Citrini, "Risultati tipici sul problema della corda vibrante con ostacolo puntiforme," I.A.C. (Istituto per le Applicazioni del Calcolo "Mauro Picone") Pubbl. Ser. III, v. 134, 1978, pp. 1-24.
  • [5] Michel Crouzeix and Alain L. Mignot, Analyse numérique des équations différentielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1984 (French). MR 762089
  • [6] G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateral constraint at the boundary, J. Differential Equations 53 (1984), no. 3, 309–361. MR 752204, https://doi.org/10.1016/0022-0396(84)90030-5
  • [7] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
  • [8] R. S. Phillips, The adjoint semi-group, Pacific J. Math. 5 (1955), 269–283. MR 0070976
  • [9] R. T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math. 39 (1971), 439–469. MR 0310612
  • [10] Michelle Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Anal. 2 (1978), no. 3, 355–373. MR 512664, https://doi.org/10.1016/0362-546X(78)90022-6
  • [11] Michelle Schatzman, Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: la corde vibrante avec obstacle ponctuel, J. Differential Equations 36 (1980), no. 2, 295–334 (French, with English summary). MR 574341, https://doi.org/10.1016/0022-0396(80)90068-6
  • [12] M. Schatzman & M. Bercovier, On the Numerical Approximation of a Vibration Problem with Unilateral Constraints, Rapport interne, Centre de Mathématiques Appliquées, Ecole Polytechnique, vol. 124, 1985.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M25, 65N30

Retrieve articles in all journals with MSC: 65M25, 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1989-0969491-5
Article copyright: © Copyright 1989 American Mathematical Society