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

Abstract

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.

Optimal Internal Dissipation of a Damped Wave Equation Using a Topological Approach

“…In (Münch and Pazoto, 2007), it is shown that this scheme is a convergent approximation of (1) under the condition Δt ≤ h/ √ 2 and provides a uniform approximation of the energy as h and Δt tend to zero. The same modification is needed for the adjoint problem.…”

“…leading to a centered scheme of order two in space and time, which is stable under the condition Δt ≤ h/ √ 2 (we refer to (Münch and Pazoto, 2007) for details). However, as observed initially in (Banks et al, 1991) in the context of stabilization and in (Glowinski et al, 1989) in the context of exact controllability, this scheme is not uniformly convergent with respect to the dissipation property.…”

“…Consequently, the time needed to uniformly damp the numerical waves from a subset of Ω\ω in which they propagate may tend to infinity as the mesh becomes finer. Thus, the dissipation mechanism may disappear completely if the initial conditions are represented by high frequency components (see (Münch and Pazoto, 2007) for numerical illustrations). This is the case for discontinuous initial data.…”

“…From a numerical viewpoint, we discuss the approximation of the wave equation in such a way that the spurious high frequencies get damped out uniformly with respect to the discretization parameters. In order to ensure the convergence of the derivative of the energy derivative (which is necessary for the convergence of the discrete optimal design), we use a modified scheme with viscosity terms introduced and analyzed in (Münch and Pazoto, 2007).…”

Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

“…Hence, damping is whether an unavoidable presence in physical reality or a desired characteristic in design. The use of advanced numerical techniques to solve the related PDEs, such as FEM and the finite difference methods (FDM) is well established and it is standard in this framework , even if the research of a numerical method that could reproduce the expected damping decay is an actual argument in literature (see [14][15][16] and their references). On the other hand, in the context of BEMs the analysis of dissipation through damped wave equation rewritten as a BIE is a relatively new topic, because it has been scarcely investigated until now.…”