Abstract: Shock-capturing schemes are capable of solving the shallow-water equations with strong discontinuities. However, since the conservative form equations are employed, the water surface gradient term is split into two terms, the gradient of pressure and the bed slope source term. Nonphysical results will be produced if the latter two terms are not well balanced. Based on theoretical analysis a novel algorithm is proposed in this paper. The non-conservative form equations are utilized inside cells in the algorithm, which retains the water surface gradient term. While the conservative form equations of the conservative variables are utilized at cell interfaces to capture discontinuities. The proposed algorithm is verified using four benchmark problems, and the results demonstrate that it can successfully treat strong discontinuities and transcritical flows over varied topography, and it can also accurately reproduce the propagation of small perturbations.