  This work focuses on the formulation of a four-equation model for simulating
unsteady two-phase mixtures with phase transition and strong discontinuities.
The main assumption consists in a homogeneous temperature, pressure and
velocity fields between the two phases.
  Specifically, we present the extension of a residual distribution scheme to
solve a four-equation two-phase system with phase transition written in a
non-conservative form, i.e. in terms of internal energy instead of the
classical total energy approach. This non-conservative formulation allows
avoiding the classical oscillations obtained by many approaches, that might
appear for the pressure profile across contact discontinuities. The proposed
method relies on a Finite Element based Residual Distribution scheme which is
designed for an explicit second-order time stepping.
  We test the non-conservative Residual Distribution scheme on several
benchmark problems and assess the results via a cross-validation with the
approximated solution obtained via a conservative approach, based on a HLLC
scheme. Furthermore, we check both methods for mesh convergence and show the
effective robustness on very severe test cases, that involve both problems with
and without phase transition.
