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 [Z. Anal. Anwend. 40 (2021)(1), 67–96] 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.

中文翻译：

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一维局部耦合波动方程稳定化的数值研究

在本文中，我们研究了由速度耦合的两个波动方程的一维系统的数值稳定性，并且内部局部控制仅作用于一个方程。在这项研究的理论部分[Z. 肛门 安文德 40（2021）（1），67–96]我们区分了两种情况。在第一个中，假定的两个波以相同的速度传播。在适当的几何条件下，我们证明了能量呈指数衰减。在第二种情况下，当波以不同的速度传播时，在适当的几何条件下，我们证明了能量仅以多项式速率衰减。在本文中，我们以一维数值逼近确认了这两个结果。然而，当耦合区域不与阻尼区域相交时，系统的稳定性在理论上仍然是一个未解决的问题。但，