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

Arnaud Münch; Ademir Fernando Pazoto

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 2, page 265-293
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topMünch, Arnaud, and Pazoto, Ademir Fernando. "Uniform stabilization of a viscous numerical approximation for a locally damped wave equation." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 265-293. <http://eudml.org/doc/250013>.

@article{Münch2007,

abstract = {
This work is devoted to the
analysis of a viscous finite-difference space semi-discretization
of a locally damped wave equation in a regular 2-D domain. The
damping term is supported in a suitable subset of the domain, so
that the energy of solutions of the damped continuous wave
equation decays exponentially to zero as time goes to infinity.
Using discrete multiplier techniques, we prove that adding a
suitable vanishing numerical viscosity term leads to a uniform
(with respect to the mesh size) exponential decay of the energy
for the solutions of the numerical scheme. The numerical viscosity
term damps out the high frequency numerical spurious oscillations
while the convergence of the scheme towards the original damped
wave equation is kept, which guarantees that the low frequencies
are damped correctly. Numerical experiments are presented and
confirm these theoretical results. These results extend those by
Tcheugoué-Tébou and Zuazua [Numer. Math.95, 563–598 (2003)] where the 1-D case
was addressed as well the square domain in 2-D. The methods and
results in this paper extend to smooth domains in any space
dimension.
},

author = {Münch, Arnaud, Pazoto, Ademir Fernando},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Wave equation; stabilization; finite difference; viscous terms; energy bound; method of lines; damped wave equation; (semi-)discretization; finite differences; convergence; numerical experiments},

language = {eng},

month = {5},

number = {2},

pages = {265-293},

publisher = {EDP Sciences},

title = {Uniform stabilization of a viscous numerical approximation for a locally damped wave equation},

url = {http://eudml.org/doc/250013},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Münch, Arnaud

AU - Pazoto, Ademir Fernando

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/5//

PB - EDP Sciences

VL - 13

IS - 2

SP - 265

EP - 293

AB -
This work is devoted to the
analysis of a viscous finite-difference space semi-discretization
of a locally damped wave equation in a regular 2-D domain. The
damping term is supported in a suitable subset of the domain, so
that the energy of solutions of the damped continuous wave
equation decays exponentially to zero as time goes to infinity.
Using discrete multiplier techniques, we prove that adding a
suitable vanishing numerical viscosity term leads to a uniform
(with respect to the mesh size) exponential decay of the energy
for the solutions of the numerical scheme. The numerical viscosity
term damps out the high frequency numerical spurious oscillations
while the convergence of the scheme towards the original damped
wave equation is kept, which guarantees that the low frequencies
are damped correctly. Numerical experiments are presented and
confirm these theoretical results. These results extend those by
Tcheugoué-Tébou and Zuazua [Numer. Math.95, 563–598 (2003)] where the 1-D case
was addressed as well the square domain in 2-D. The methods and
results in this paper extend to smooth domains in any space
dimension.

LA - eng

KW - Wave equation; stabilization; finite difference; viscous terms; energy bound; method of lines; damped wave equation; (semi-)discretization; finite differences; convergence; numerical experiments

UR - http://eudml.org/doc/250013

ER -

## References

top- M. Asch and G. Lebeau, The spectrum of the damped wave operator for geometrically complex domain in ${\mathbb{R}}^{2}$. Experimental Math.12 (2003) 227–241. Zbl1061.35064
- H.T. Banks, K. Ito and B. Wang, Exponentially stable approximations of weakly damped wave equations. Ser. Num. Math.100 Birkhäuser (1990) 1–33. Zbl0850.93719
- C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Opt.30 (1992) 1024–1065. Zbl0786.93009
- D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl.52 (1975) 189–219. Zbl0317.49005
- G.C. Cohen, Higher-order Numerical Methods for Transient Wave Equations. Scientific Computation, Springer (2002). Zbl0985.65096
- C.M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal.29 (1968) 241–271. Zbl0183.37701
- R. Glowinski, C.H. Li and J.-L. Lions, 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. Zbl0699.65055
- A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations59 (1985) 145–154. Zbl0535.35006
- A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portug. Math.46 (1989) 245–258. Zbl0679.93063
- A. Henrot, Continuity with respect to the domain for the Laplacian: a survey. Control Cybernetics23 (1994) 427–443. Zbl0822.35029
- J.A. Infante and E. Zuazua, Boundary observability for the space-discretizations of the 1-D wave equation. ESAIM: M2AN33 (1999) 407–438. Zbl0947.65101
- V. Komornik, Exact Controllability and Stabilization - The Multiplier Method. J. Wiley and Masson (1994). Zbl0937.93003
- S. Krenk, Dispersion-corrected explicit integration of the wave equation. Comput. Methods Appl. Mech. Engrg.191 (2001) 975–987. Zbl1009.76054
- J. Lagnese, Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt.21 (1983) 68–85. Zbl0512.93014
- 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). Zbl0165.10801
- A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN39 (2005) 377–418. Zbl1130.93016
- M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math.95 (1996) 25–42. Zbl0860.35072
- M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. C.R. Acad. Sci. Paris338 (2004) 281–286. Zbl1040.93030
- O. Pironneau, Optimal shape design for elliptic systems. New York, Springer (1984). Zbl0534.49001
- 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). Zbl1126.93050
- M. Slemrod, Weak asymptotic decay via a “Relaxed Invariance Principle” for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh113 (1989) 87–97. Zbl0699.35023
- L.R. Tcheugoué-Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations145 (1998) 502–524. Zbl0916.35069
- L.R. Tcheugoué-Tébou and E. Zuazua, 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. Zbl1033.65080
- E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equation15 (1990) 205–235. Zbl0716.35010
- E. Zuazua, 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. Zbl0939.93016
- E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation. Rendiconti di Matematica, Serie VIII24 (2004) 201–237. Zbl1085.49041
- E. Zuazua, Propagation, observation, control and numerical approximation of waves. SIAM Rev.47 (2005) 197–243. Zbl1077.65095

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.