# 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

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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 -

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