We have conducted a study where we applied one-dimensional central-Upwind methods to estimate solutions of the Saint-Venant (SV) system. Our approach carefully considers the source terms associated with bottom topography. Within the context of the SV system, there are steady-state solutions that arise when the non-zero gradients of flux are exactly balanced by the corresponding source terms. Maintaining this delicate equilibrium with numerical approaches presents a challenging problem. Finding slight variations in these states proves to be extremely difficult in the field of computing. To address this, we propose an extension of semi-discrete central schemes, commonly employed in hyperbolic conservation law systems, to encompass balance laws (BL). In our approach, we specifically focus on discretizing the source term with great care and attention. To verify the superior accuracy, precise preservation of the C-property, and outstanding resolution of our approach, we perform comprehensive one-dimensional simulations on both continuous and discontinuous solutions.