We present one- and two-dimensional central-upwind schemesfor approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutionsin which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preservethis delicate balance with numerical schemes.Small perturbations of these states are also very difficultto compute. Our approach is based on extending semi-discrete central schemes forsystems of hyperbolic conservation laws to balance laws.Special attention is paid to the discretization of the sourceterm such as to preserve stationary steady-statesolutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water.This important feature allows one to compute solutions for problemsthat include dry areas.
