7 This work aims at developing a high-order numerical method for the propagation of acoustic shock waves using the discontinuous Galerkin method. High order methods tend to amplify the formation of spurious oscillations (Gibbs phenomenon) around the discontinuities/shocks, associated to the relative importance of higher-harmonics resulting from nonlinear propagation (in our case). To handle this critical issue, a new shock sensor is introduced for the sub-cell shock capturing. Thereafter, an element-centered smooth artificial viscosity is introduced into the system wherever an acoustic shock wave is sensed. Validation tests in 1D and 2D configurations show that the method is well-suited for the propagation of acoustic shock waves along with other physical effects like geometrical spreading and diffraction.