Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

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 semidiscretization 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 − d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes 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.

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

“…If we are interested in the observability inequality (1.12) for a particular subinterval (a, b) ⊂ (0, 1), the situation is more intricate. As above, due to the explicit description of the energies (2.17) and (2.18), one easily checks that if there exists a constant M 3 such that for all n ∈ N, 28) then for all n ∈ N and for all k ∈ {1, . .…”

“…To our knowledge, this question has not been addressed so far. We expect this question to be difficult to address with the tools used until now, which require either a good knowledge of the eigenvalues (see [5,6,17,[23][24][25][26]31] and our own approach) or the existence of multipliers that behave well (see [10,27,28]) on the discrete systems. This issue is currently under investigation by the author.…”

“…A new approach was proposed recently in [25] based on wavelet filtering techniques. Let us also mention the results [10,[26][27][28] that amounts to adding an extra term in (1.12) which is non-negligible only for the high frequencies. A last possible cure was proposed in [1,15] and later analyzed in [5]: a 1d semi-discrete scheme derived from a mixed finite element method was proposed, which has the property that the group velocity of the waves is bounded from below.…”

Approximation of periodic solutions for a dissipative hyperbolic equation

“…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))…”

“…It is known that this procedure ensures the uniform exponential decay of (1.6) (see, for instance, [12,13,19,21,24,25]). More precisely, the solution (u h ,u h ) of (1.6) verifies (1.5) with constants M and ω independent of h. Since the term h η A huh (t) enforces the dissipation, the exponential decay of the solutions of (1.6) gets better when η decreases.…”