A high-order numerical method is developed for the computation of compressible multiphase flows. The model we use is based on the Baer \& Nunziato type systems [M.R. Baer, J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials, Int. J. Multiphase Flows, 12 (12) (1986), pp. 861-889]. Among all the other available models in the literature, these systems present the advantage to be able to simulate either interface or mixture problems. Nevertheless, they still raise some issues, mainly based on their non-conservative feature. The numerical method we propose is a discontinuous Galerkin type. In this work, the interior side integrals are computed thanks to [R. Abgrall, R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures, J. Comput. Phys., 186 (361-396) (2003)]. Robustness and high order of accuracy of the method are proved on classical interface problems, but also on suitably derived analytical solutions.