We propose a high order diffuse interface method for dealing with compressible multiphase flows with interfaces. This scheme is based on the discontinuous Galerkin formulation of [E. Franquet, V. Perrier, Runge-Kutta Discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models, Journal of Computational Physics (2012)]. As it is linear, this scheme is oscillating, so that the volume fraction can become negative or greater than 1. For stabilizing it, the maximum preserving limiter introduced in [X. Zhang, Y. Xia, C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for con- servation laws on triangular meshes, Journal of Scientific Computing. (2012)] is used. The scheme is applied to the computation of a Rayleigh-Taylor instability.