This paper describes progress on a consistent approach for multi-phase flow modelling with phase-change. Although, the de- veloped methods are general purpose the applications presented here cover the LIVE experiments involving core melt phenomena at the lower vessel head of a nuclear reactor. These include corium pool formation, coolability and solidification. A new method for solving the compositional multi-phase flow equations to calculate material indicator fields is adopted. An interface-capturing scheme based on high-order accurate compressive advection methods including a Petrov-Galerkin approach is employed to main- tain sharpness of the interfaces between materials. A novel control volume-finite element mixed formulation based on the P1DG-P2 (linear discontinuous in velocity and quadratic continuous in pressure) element pair which can uniquely ensure that key balances are maintained in the equations is developed to discretise the governing equations in space. Anisotropic mesh adaptivity is used to focus the numerical resolution around the interfaces and other areas of important dynamics.