In this paper, a numerical solver is developed for the computation of one and two dimensional dam break problems. The considered equations are the 2D shallow water equations written in conservative form. The algorithm uses a finite volume method which is based on Roe’s approximate Riemann solver. It is of second order in space and time, and can be used on complicated geometries with unstructured meshes. The stiffness coming from discontinuity propagation due to the dam is taken into account by the introduction of a dynamical mesh refinement-unrefinement procedure. The results presented on some benchmark dam break situations including wet/dry beds, and comparisons with analytical solutions, show the accuracy of the used methods and the efficiency of the adaptation technique in the simulation of such phenomena.