Lagrangian block advection is introduced as an alternative numerical method to the classical method of the Eulerian flux. The method of the block advection is positive-depth definite and free of numerical oscillation. In simulation of waves in shallow water, it prevents the water depth from become negative at the wet-and-dry interface. The method also does not produce numerical oscillation across breaking waves. Simulations of dam-break flood waves and bores runup on steep slopes are carried out using the Lagrangian block advection method. Grid refinement studies are conducted to show the computational accuracy and stability of the method.