Phase transition in compressible flows involves capillarity effects, described by the Euler-Korteweg equations with non-convex equation of state. Far from phase transition, i.e. in the two convex parts of the equation of state, the dispersion terms vanish and one should recover the hyperbolic Euler equations of fluid dynamics. However, the solution of Euler-Korteweg equations does not converge towards the solution of Euler equations when dispersion tends towards zero while being non-null: it is a singular limit problem. To avoid this issue in the case of convex equation of state, a Navier-Stokes-Korteweg model is considered, whose viscosity is chosen to counterbalance exactly the dispersive terms. In the limit of small viscosity and small dispersion, the Euler model is recovered. Numerically, an extended Lagrangian method is used to integrate the Navier-Stokes-Korteweg equations so-obtained. Doing so allows to use classical numerical schemes of Godunov type with source term. Numerical results for a Riemann problem illustrate the convergence properties with vanishing dispersion