Fluid flows are very often governed by the dynamics of a small number of coherent structures, i.e., fluid features which keep their individuality during the evolution of the flow. The purpose of this paper is to present a way to simulate the Euler equations on the basis of the evolution of such coherent structures. One way to extract from flow simulations some basis functions which can be interpreted as coherent structures is by Proper Orthogonal Decomposition (POD). Then, by means of a Galerkin projection, it is possible to find the system of ODEs which approximates the problem in the finite dimensional space spanned by the basis functions found. Issues concerning the stability and the accuracy of such an approximation are discussed. It is found that a straight-forward Galerkin method is unstable. Some features of discontinuous Galerkin methods are therefore incorporated to achieve stability, which is proved for a linear scalar case. In addition, we propose a way to reduce the cost of the computation and to increase accuracy at the same time. Some one-dimensional computational experiments are presented, including shock tube simulations and rarefaction fans
