# Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach

Marica A, Zuazua E (2014)

**Publication Type:** Book chapter / Article in edited volumes

**Publication year:** 2014

### Journal

**Publisher:** Springer Verlag

**Edited Volumes:** Optimization with PDE Constraints

**Series:** Lecture Notes in Computational Science and Engineering

**Book Volume:** 101

**Pages Range:** 285-324

**DOI:** 10.1007/978-3-319-08025-3_9

### 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 (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on somenon-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 Tébou 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.

### Authors with CRIS profile

### Involved external institutions

### How to cite

**APA:**

Marica, A., & Zuazua, E. (2014). Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach. In Ronald Hoppe (Eds.), *Optimization with PDE Constraints.* (pp. 285-324). Springer Verlag.

**MLA:**

Marica, Aurora, and Enrique Zuazua. "Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach." *Optimization with PDE Constraints.* Ed. Ronald Hoppe, Springer Verlag, 2014. 285-324.

**BibTeX:** Download