Hyperbolic systems and dispersive equations remain challenging for the FEM community. On the basis of an arbitrarily high order FEM, namely the spectral element method (SEM), here we address: - The Korteweg-de Vries equation, to explain how high order derivative terms can be efficiently handled with a C 0 continuous Galerkin approximation. Two strategies are proposed, both of them allowing the SEM approximation of the high order derivative term to remain in the usual H1 space. The conservation of the invariants is also focused on, especially by using in time embedded implicit-explicit Runge Kutta schemes [1]. - The 2D shallow water equations, to show how a stabilized SEM can solve problems involving shocks. Moreover, we especially focus on flows involving dry-wet transitions and propose to this end an efficient variant of the entropy viscosity method [2, 3]. Results obtained for well known benchmark problems are provided to illustrate the capabilities of the proposed high order algorithms.