Numerical study of the stabilization of 1D locally coupled wave equations

TL;DR

This paper numerically investigates the stabilization of a 1D coupled wave system, confirming theoretical decay rates and revealing unexpected behaviors when coupling and damping regions do not intersect.

Contribution

It provides numerical validation of theoretical decay rates for coupled wave equations and uncovers new behaviors in stabilization when coupling regions are disjoint from damping regions.

Findings

Energy decays exponentially when waves propagate at the same speed

Energy decays polynomially when waves propagate at different speeds

Unpredicted exponential decay observed even when coupling and damping regions do not intersect

Abstract

In this paper, we study the numerical stabilization of a 1D system of two wave equations coupled by velocities with an internal, local control acting on only one equation. In the theoretical part of this study, we distinguished two cases. In the first one, the two waves assumed propagate at the same speed. Under appropriate geometric conditions, we had proved that the energy decays exponentially. While in the second case, when the waves propagate at different speeds, under appropriate geometric conditions, we had proved that the energy decays only at a polynomial rate. In this paper, we confirmed these two results in a 1D numerical approximation. However, when the coupling region does not intersect the damping region, the stabilization of the system is still theoretically an open problem. But, here in both cases, we observed an unpredicted behavior : the energy decays at an exponential rate when the propagation speeds are the same or at a polynomial rate when they are different.