  This paper presents a high-order discontinuous Galerkin finite element method
to solve the barotropic version of the conservative symmetric hyperbolic and
thermodynamically compatible (SHTC) model of compressible two-phase flow,
introduced by Romenski et al., in multiple space dimensions. In the absence of
algebraic source terms, the model is endowed with a curl constraint on the
relative velocity field. In this paper, the hyperbolicity of the system is
studied for the first time in the multidimensional case, showing that the
original model is only weakly hyperbolic in multiple space dimensions. To
restore strong hyperbolicity, two different methodologies are used: i) the
explicit symmetrization of the system, which can be achieved by adding terms
that contain linear combinations of the curl involution, similar to the
Godunov-Powell terms in the MHD equations; ii) the use of the hyperbolic
generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The
PDE system is solved using a high-order ADER discontinuous Galerkin method with
a posteriori sub-cell finite volume limiter to deal with shock waves and the
steep gradients in the volume fraction commonly appearing in the solutions of
this type of model. To illustrate the performance of the method, several
different test cases and benchmark problems have been run, showing the
high-order of the scheme and the good agreement when compared to reference
solutions computed with other well-known methods.
