Volume 13, Number 2, April-June 2007
|Page(s)||265 - 293|
|Published online||12 May 2007|
Uniform stabilization of a viscous numerical approximation for a locally damped wave equation
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; firstname.lastname@example.org
2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, 21940-970, Rio de Janeiro, Brasil; email@example.com
Revised: 10 August 2005
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.
Mathematics Subject Classification: 65M06
Key words: Wave equation / stabilization / finite difference / viscous terms
© EDP Sciences, SMAI, 2007
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.