<p>We present a conservative level set (CLS) method variant with provable boundedness properties for interface capturing. A central finite difference scheme is implemented to discretize the governing equations of the modified CLS method, in which a small diffusion term is added into the advection step of the original CLS method to reduce the numerical oscillations near the interface. By simply controlling the diffusion during the advection and reinitialization steps, we prove that the boundedness of the level set (LS) function can be guaranteed without any extra complex nonlinear limiter or mass redistribution technique for the Euler forward scheme and high-order Runge–Kutta (RK) time integration schemes. Several standard advection benchmarks for interface capturing are presented to verify the performance of our proposed method, i.e., mass conservation, boundedness, and accuracy.</p>