Hyperbolic systems and dispersive equations remain challenging for finite element methods (FEMs). On the basis of an arbitrarily high order FEM, namely the spectral element method (SEM), we address : -The Korteweg-de Vries equation, to explain how high order derivative terms can be efficiently handled with a C0-continuous Galerkin approximation. The conservation of the invariants is also focused on, especially by using in time embedded implicit-explicit Runge Kutta schemes. -The 2D shallow water equations, to show how a stabilized SEM can solve problems involving shocks. We especially focus on flows involving dry-wet transitions and propose to this end an efficient variant of the entropy viscosity method.