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- Title
Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation.
- Authors
Louis Tebou; Enrique Zuazua
- Abstract
Abstract??The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?dLaplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing sizeh. Actually, the decay rate tends to zero ashgoes to zero. Then 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 of solutions. This 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. Our method of proof relies essentially on discrete multiplier techniques.
- Subjects
FINITE differences; NUMERICAL analysis; WAVE equation; OSCILLATIONS
- Publication
Advances in Computational Mathematics, 2007, Vol 26, Issue 1-3, p337
- ISSN
1019-7168
- Publication type
Article
- DOI
10.1007/s10444-004-7629-9