Quasi exponential de ay of a nite dieren e
29 Pages
English
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Quasi exponential de ay of a nite dieren e

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29 Pages
English

Description

Niveau: Secondaire, Lycée, Terminale
Quasi exponential de ay of a nite dieren e spa e dis retization of the 1-d wave equation by pointwise interior stabilization Serge Ni aise, Julie Valein Université de Valen iennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956 Institut des S ien es et Te hniques of Valen iennes F-59313 - Valen iennes Cedex 9 Fran e Julie.Valein,Serge.Ni aiseuniv-valen iennes.fr O tober 7, 2008 Abstra t We onsider the wave equation on an interval of length 1 with an interior damping at ?. It is well-known that this system is well-posed in the energy spa e and that its natural energy is dissipative. Moreover, as it was proved in [1?, the exponential de ay property of its solution is equivalent to an observability estimate for the orresponding onservative system. In this ase, the observability estimate holds if and only if ? is a rational number with an irredu tible fra tion ? = p q , where p is odd, and therefore under this ondition, this system is exponentially stable in the energy spa e. In this work, we are interested in the nite dieren e spa e semi- dis retization of the above system. As for other problems [24, 21?, we an expe t that the exponential de ay of this s heme does not hold in general due to high frequen y spurious modes.

  • tion

  • without any proof

  • y??j ?

  • tion ?

  • interior damp- ing

  • pointwise damping

  • numeri al


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21
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y(ξ , t) =y(ξ , t) t> 0,− +
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N+1
(0, 1)
0 =x <x =h<... <x <x =jh<x <...<x <x = 1,0 1 j−1 j j+1 N N+1
x =jh j = 0,..., N +1. ξ jhj
j, j ∈N∩(0, N +1) x →ξ N →∞.N jN

y −2y +y′′ j+1 j j−1 y − = 0 t> 0, j = 1,..., N, j =j ,2 N j h y = 0, y −y = 0 t> 0,0 N+1 N
y −2y +yj +1 j j −1 ′N N N =αy t> 0,j h N (0) (1)′y (t = 0) =y , y (t = 0) =y j = 1,..., N.j j j j
y (t) y(x , t), yj j
(0) (1)
(y , y ), j = 0,..., N +1j j
y −2y +yj +1 j j −1 ′N N N = αyjh N
y (ξ , t)−y (ξ , t) =−αy (ξ, t).x − x + t
y −2y +yj +1 j j −1′′ N N N ′hy − =−αy ,j jN Nh
(0) (0) (1) (1)
y = (y ) , y = (y ) y = (y ) .h j j j jh j h j

′v =A vh hh ,
v (0) =vh h0
! (0)
y yh hv = v = A 2N×2Nh h0 h′ (1)y yh h
N N 2X X h h y (t)−y (t)2 j+1 j′