Boundary observability for the space-discretizations of the 1 — d wave equation

Río, Juan Antonio Infante del; Zuazua, Enrique

doi:10.1016/s0764-4442(98)80036-0

Cited by 40 publications

(63 citation statements)

References 3 publications

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“…In the case F = 0, it is known that (2.15)-(2.16) defines a well-posed dynamical system in the space X. More precisely, for each U 0 ∈ X, the solution U of (2.15)-(2.16) is given by 17) where S is the contraction semigroup on X generated by the unbounded operator A in…”

Section: Existence Of Periodic Solutionsmentioning

confidence: 99%

“…When considering numerical discretization schemes for wave equations, it is well known that most of them do not preserve the uniform (with respect to the mesh-size h) decay property of the solutions of the continuous wave equation (1.2). Indeed, as remarked in [24,25] (see, also, [4,17,27,28]), due to the existence of high frequency spurious solutions whose (group) velocity of propagation is of the order of h, the energy of the discrete solution (u h ,u h ) does not have a uniform exponential decay. This means that the discrete energy of solutions defined by E h (t) = 1 2 (u h (t),u h (t))…”

Section: Introductionmentioning

confidence: 96%

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Approximation of periodic solutions for a dissipative hyperbolic equation

2013

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This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force f . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon's consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on f , the approximation method is always convergent. However, our error estimates show that the convergence's properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term f is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.Mathematics Subject Classification (2010): 35B10, 65P99, 93C20

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Section: Existence Of Periodic Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Approximation of periodic solutions for a dissipative hyperbolic equation

2013

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“…This fact may be a strong limitation if we know how an additional discretization in time could perturb properties of a scheme. Thus, at the semi-discrete level, it was shown in [12,26] that the filtering of the high frequency modes leads to a uniform observability inequality for the adjoint system. This is equivalent to the uniform controllability of the projection of the solution over the space generated by the remaining eigenmodes.…”

Section: Introductionmentioning

confidence: 99%

A uniformly controllable and implicit scheme for the 1-D wave equation

Münch¹

2005

ESAIM: M2AN

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Abstract. This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and ∆t. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h 2 and ∆t 2 . Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L 2 (0, T ) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L 2 -norm control. The results are illustrated with several numerical experiments.Mathematics Subject Classification. 35L05, 65M60, 93B05.

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“…1). Then, we associate to the first equation of (1) the finite-difference semi-discretization equation (11) where y j,k (t) denotes the approximation of y(., t) at the node (jh 1 , kh 2 ) of Ω h1,h2 . Following the methodology developed in [24], we show the efficiency of this scheme : we first prove that the energy associated with this numerical scheme decays exponentially, with a rate independent of the mesh size (see Th.…”

mentioning

confidence: 99%

“…To obtain the result we combine discrete multiplier methods introduced in [11] and compactness-uniqueness arguments. Let us note that a discrete version of multipliers was also developed in [11,26] to address the issue of boundary observability in 1-D and the 2-D square domain. Then, we prove that the solution of the discretized system defined on Ω h1,h2 converges to the solution of (1) defined on Ω (see Th.…”

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confidence: 99%

Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

Münch

Pazoto

2007

ESAIM: COCV

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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.Mathematics Subject Classification. 65M06.

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Boundary observability for the space-discretizations of the 1 — d wave equation

Cited by 40 publications

References 3 publications

Approximation of periodic solutions for a dissipative hyperbolic equation

Approximation of periodic solutions for a dissipative hyperbolic equation

A uniformly controllable and implicit scheme for the 1-D wave equation

Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

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