Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T00:23:04.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

Published online by Cambridge University Press:  12 May 2007

Arnaud Münch  and
Ademir Fernando Pazoto
Arnaud Münch
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, UFR de Sciences et Techniques, Université de Franche-Comté, 16, route de Gray 25030, Besançon Cedex, France; arnaud.munch@univ-fcomte.fr
Ademir Fernando Pazoto
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, 21940-970, Rio de Janeiro, Brasil; ademir@acd.ufrj.br
Get access

Abstract

This work is devoted to theanalysis of a viscous finite-difference space semi-discretizationof a locally damped wave equation in a regular 2-D domain. Thedamping term is supported in a suitable subset of the domain, sothat the energy of solutions of the damped continuous waveequation decays exponentially to zero as time goes to infinity.Using discrete multiplier techniques, we prove that adding asuitable vanishing numerical viscosity term leads to a uniform(with respect to the mesh size) exponential decay of the energyfor the solutions of the numerical scheme. The numerical viscosityterm damps out the high frequency numerical spurious oscillationswhile the convergence of the scheme towards the original dampedwave equation is kept, which guarantees that the low frequenciesare damped correctly. Numerical experiments are presented andconfirm these theoretical results. These results extend those byTcheugoué-Tébou and Zuazua [Numer. Math.95, 563–598 (2003)] where the 1-D casewas addressed as well the square domain in 2-D. The methods andresults in this paper extend to smooth domains in any spacedimension.

Keywords

Wave equationstabilizationfinite differenceviscous terms
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 265 - 293
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asch, M. and Lebeau, G., The spectrum of the damped wave operator for geometrically complex domain in $\mathbb{R}^2$ . Experimental Math. 12 (2003) 227–241. CrossRef
H.T. Banks, K. Ito and B. Wang, Exponentially stable approximations of weakly damped wave equations. Ser. Num. Math. 100 Birkhäuser (1990) 1–33.
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Opt. 30 (1992) 1024–1065. CrossRef
Chenais, D., On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189–219. CrossRef
G.C. Cohen, Higher-order Numerical Methods for Transient Wave Equations. Scientific Computation, Springer (2002).
Dafermos, C.M., On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29 (1968) 241–271. CrossRef
Glowinski, R., Li, C.H. and Lions, J.-L., A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet Controls: Description of the numerical methods. Japan. J. Appl. Math. 7 (1990) 1–76. CrossRef
Haraux, A., Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985) 145–154. CrossRef
Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portug. Math. 46 (1989) 245–258.
Henrot, A., Continuity with respect to the domain for the Laplacian: a survey. Control Cybernetics 23 (1994) 427–443.
Infante, J.A. and Zuazua, E., Boundary observability for the space-discretizations of the 1-D wave equation. ESAIM: M2AN 33 (1999) 407–438. CrossRef
V. Komornik, Exact Controllability and Stabilization - The Multiplier Method. J. Wiley and Masson (1994).
Krenk, S., Dispersion-corrected explicit integration of the wave equation. Comput. Methods Appl. Mech. Engrg. 191 (2001) 975–987. CrossRef
Lagnese, J., Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt. 21 (1983) 68–85. CrossRef
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969).
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).
Münch, A., A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377–418. CrossRef
Nakao, M., Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25–42. CrossRef
Negreanu, M. and Zuazua, E., Discrete Ingham inequalities and applications. C.R. Acad. Sci. Paris 338 (2004) 281–286. CrossRef
O. Pironneau, Optimal shape design for elliptic systems. New York, Springer (1984).
K. Ramdani, T. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations: Application to the optimal controle of flexible structures. Technical report, Prépublications de l'Institut Elie Cartan 27 (2003).
Slemrod, M., Weak asymptotic decay via a “Relaxed Invariance Principle” for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh 113 (1989) 87–97. CrossRef
Tcheugoué-Tébou, L.R., Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations 145 (1998) 502–524. CrossRef
Tcheugoué-Tébou, L.R. and Zuazua, E., Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95 (2003) 563–598. CrossRef
Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equation 15 (1990) 205–235.
Zuazua, E., Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523–563. CrossRef
E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation. Rendiconti di Matematica, Serie VIII 24 (2004) 201–237.
Zuazua, E., Propagation, observation, control and numerical approximation of waves. SIAM Rev. 47 (2005) 197–243. CrossRef