ABSTRACTThis paper report progress on a technique to stabilize the consistent mass SUPG method, when it is used in combination with an explicit scheme to solve shallow water ow problems to steady solutions. According to this a comparison between the use of lumped mass SUPG and consistent mass SUPG method is done. A scale of di usivity in this scheme is developed by means of the introduction of parameter in the mass matrix of the consistent SUPG method and a stability analysis of -SUPG method is presented. Several examples based on the one-dimensional shallow-water equations illustrate the accuracy and e ciency obtained with such methods.KEYWORDS:  shallow water; steady solutions; parametric method.