Abstract: The propagation of gaseous detonation waves under radially diverging conditions is commonly encountered in rotating detonation engines and industrial explosion safety issues. The study of their propagation stability holds significant value in both theoretical exploration and engineering applications. This paper quantifies the radial divergence effect using the detonation wavefront curvature as a parameter and establishes computational model for one-dimensional detonation wave propagation with radial divergence, based on a one-step irreversible chemical reaction. One-dimensional nonlinear stability of the detonation waves is investigated by using the numerical simulations with high-resolution scheme. The effects of activation energy and wavefront curvature on the detonation wave stability are examined, and the critical conditions of stability and quenching during detonation wave propagation are revealed. The results indicate that for the reactive system with specific heat ratio of γ = 1.2 and chemical heat release of Q = 50RT₀, increasing curvature can cause the detonation wave to transition directly from stable propagation mode to quenching mode when activation energy is less than 22.3RT₀. For moderate activation energies ranged from 22.3RT₀ to 25.3RT₀, the increase in curvature leads to a shift from stable propagation to periodic oscillation mode, followed by irregular oscillation mode and eventual quenching. When the activation energy is larger than 25.3RT₀, raising curvature can make the detonation wave change from irregular oscillation mode to quenching. A modified stability parameter χ for the detonation waves considering the radially diverging is proposed. Further analysis using this modified parameter shows that the range of χ = 1.2 ~ 1.35 aligns well with the stability boundary observed in numerical simulations. Additionally, the results obtained in present study are compared with predictions from linear stability theory in existing literature. The good consistency between them demonstrates that both linear and nonlinear stability methods predict the same critical stability boundary for one-dimensional detonation wave propagation under radially diverging conditions.