  The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface
model for compressible two-phase flow. However, the numerical solution faces
two major challenges: due to a third-order dispersion contribution in the
momentum equations, extended numerical stencils are required for the flux
calculation. Furthermore, the equation of state given by a Van-der-Waals law,
exhibits non-monotone behaviour in the two-phase state space leading to
imaginary eigenvalues of the Jacobian of the first-order fluxes. In this work a
lower-order parabolic relaxation model is used to approximate solutions of the
classical NSK equations. Whereas an additional parabolic evolution equation for
the relaxation variable has to be solved, the system involves no differential
operator of higher as second order. The use of a modified pressure function
guarantees that the first-order fluxes remain hyperbolic. Altogether, the
relaxation system is directly accessible for standard compressible flow
solvers. We use the higher-order Discontinuous Galerkin spectral element method
as realized in the open source code FLEXI. The relaxation model is validated
against solutions of the original NSK model for a variety of 1D and 2D test
cases. Three-dimensional simulations of head-on droplet collisions for a range
of different collision Weber numbers underline the capability of the approach.
