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

Abstract

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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.

How to cite

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Münch, 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

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