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Article Dans Une Revue Advances in Computational Mathematics Année : 2010

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

Résumé

We consider the wave equation on an interval of length 1 with an interior damping at ξ . It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3-4):215-240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction ξ = p q , where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197-243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337-365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.

Dates et versions

hal-00599626 , version 1 (10-06-2011)

Identifiants

Citer

Serge Nicaise, Julie Valein. Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization. Advances in Computational Mathematics, 2010, 32 (3), pp.303-334. ⟨10.1007/s10444-008-9108-1⟩. ⟨hal-00599626⟩
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