The Navier-Stokes-Korteweg equations (a viscous-dispersive regularisation of the Euler equations) are considered for modelling a liquid-vapour phase transition [1, 2]. A cubic van der Waals equation of state is used to close the system. The viscosity is chosen so that the solutions of the Euler equations are recovered in the viscous-dispersiveless limit, yielding classical shocks [3]. Using an extended Lagrangian method [4], the dispersive Navier–Stokes–Korteweg system of equations is approximated by a first-order system of conservation laws. Despite the fact that the equation of state is non-convex and non-monotone, the extended system can be made unconditionally hyperbolic by an adequate choice of the extended parameters. Using a splitting method, the extended system is solved numerically by a second-order finite-volume numerical scheme. Numerical results involving non-classic waves, such as undercompressive shocks and composite waves, will be presented. [1] Korteweg D. J., Sur la forme que prennent les équation de mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité, Archives Néerlandaises des Sciences Exactes et Naturelles, 6, 1-27, 1901. [2] Van der Waals J. D., Théorie thermodynamique de la capillarité dans l’hypothèse d’une variation continue de densité, Archives Néerlandaises des Sciences Exactes et Naturelles, 8, 121-209, 1895. [3] Didierlaurent Q., Favrie N., Lombard B., Solving a Singular Limit Problem Arising With Eu-ler–Korteweg Dispersive Waves, Studies in Applied Mathematics, 154, e70005, 2025. [4] Favrie N. and Gavrilyuk S.L., A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves, Nonlinearity, 30, 2718, 2017.