This paper investigates the stability of both the semi-discrete and the implicit central scheme for the linear damped wave equation on the half-line, where the spatial boundary is characteristic for the limiting equation. The proposed schemes incorporate a discrete boundary condition designed to guarantee the uniform stability of the IBVP, regardless of the stiffness of the source term or the spatial step size. Stability estimates for the semi-discrete scheme are established using the summation-by-parts (SBP) and simultaneous-approximation-term (SAT) penalty techniques, building on the continuous framework analyzed by Xin and Xu (2000).