The Lagrangian Blocks on Eulerian Mesh (LBEM) method is developed for numerical simulation of waves in shallow water. The method is positive-depth definite and free of numerical oscillation. It prevents the water depth from become negative in repeated wetting and drying simulations. It does not produce numerical oscillation across breaking shock waves. Grid refinement studies are conducted for the wetting and drying by water in a parabolic bowl and the oblique dam-break waves. A series of flood routings in urban settings is presented as application examples in the paper taking full advantage of the computational stability of the method.