The mixture of gas and solid particles in non-equilibrium conditions that compose a multiphase flows is described with a set of coupled partial differential equations for the mass, momentum and energy of each species. Solid particles and the gas phase are considered as interpenetrating continua, following an Eulerian-Eulerian approach. Each species is compressible and inviscid. The gas and particles dynamics are coupled through the drag term in the momentum equations and the heat exchange term in the energy equations. Following the methods of lines, a p-adaptive Discontinuous Galerkin space discretization is introduced with explicit time discretization schemes. A monotonization technique is introduced on the advective terms of the system. An automatic criterion is introduced to adapt the local number of degrees of freedom and to improve the accuracy locally. The employed technique is simple and relies on the use of orthogonal hierarchical basis functions. The numerical model is applied and tested to several relevant 1D test cases, with special focus on pyroclastic flows arising in volcanic eruptions, in order to assess its accuracy and stability properties. Comparisons between different numerical approaches are presented. Moreover we analyze the efficiency of the p-adaptivity approach.