<p>In this paper, the authors analyz the uniform exponential stability of a semi-discrete scheme for a coupled system derived from a one-dimensional wave equation, which is subject to boundary feedback with noncollocated observation. This system was previously studied in the paper (Guo B Z and Xu C Z, 2007), where the Riesz basis methodology was utilized. However, it is critical to acknowledge that the Riesz basis approach is inadequate for addressing the uniform exponential stability of discrete schemes. In contrast, the Lyapunov function offers a more straightforward alternative approach. Therefore, the authors first establish exponential stability by constructing a global Lyapunov function for the closed-loop system. Subsequently, employing the order reduction method, the authors derive the semi-discrete finite difference (FD) scheme of the system. Analogous to the demonstration for the continuous case, the authors construct discrete Lyapunov functions and utilize them to demonstrate that the discretized scheme exhibits uniformly exponential decay as the step size approaches zero.</p>