  In this work we propose a high-order discretization of the Baer-Nunziato
two-phase flow model (Baer and Nunziato, Int. J. Multiphase Flow, 12 (1986),
pp. 861-889) with closures for interface velocity and pressure adapted to the
treatment of discontinuous solutions, and stiffened gas equations of states. We
use the discontinuous Galerkin spectral element method (DGSEM), based on
collocation of quadrature and interpolation points (Kopriva and Gassner, J.
Sci. Comput., 44 (2010), pp. 136-155). The DGSEM uses summation-by-parts (SBP)
operators in the numerical quadrature for approximating the integrals over
discretization elements (Carpenter et al., SIAM J. Sci. Comput., 36 (2014), pp.
B835-B867; Gassner et al., J. Comput. Phys., 327 (2016), pp. 39-66). Here, we
build upon the framework provided in (F. Renac, J. Comput. Phys., 382 (2019),
pp. 1-36) for nonconservative hyperbolic systems to modify the integration over
cell elements using the SBP operators and replace the physical fluxes with
entropy conservative fluctuation fluxes from Castro et al. (SIAM J. Numer.
Anal., 51 (2013), pp. 1371-1391), while we derive entropy stable fluxes applied
at interfaces. This allows to establish a semi-discrete inequality for the
cell-averaged physical entropy, while being high-order accurate. The design of
the numerical fluxes also formally preserves the kinetic energy at the
semi-discrete level. High-order integration in time is performed using strong
stability-preserving Runge-Kutta schemes and we propose conditions on the
numerical parameters for the positivity of the cell-averaged void fraction and
partial densities. The positivity of the cell-averaged solution is extended to
nodal values by the use of an a posteriori limiter. The high-order accuracy,
nonlinear stability, and robustness of the present scheme are assessed through
several numerical experiments in one and two space dimensions.
