My work consists in developing some high order numerical schemes in time and space for the modeling of the wave propagation. We have proposed to first discretize the wave equation with respect to the time using the so called Modified Equation Technique. Then, we have used a Discontinuous Galerkin Finite Element method for the space discretization. Switching the classical discretization process, we have constructed schemes as accurate as the classical ones with a numerical cost very interesting. After the numerical validation of this method, we have focused on its stability and on its adaptativity in time and space. To reach these objectives, we have performed a stability analysis of the Discontinuous Galerkin method and we have proposed some improvements to this technique which implie very important gain in terms of computationnal time.
