Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

Abstract

In this paper, we consider the boundary stabilization problem associated to the 1 − d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tebou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on some non-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tebou and Zuazua (Adv. Comput. Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients.