Phase change in solidification and melting can be described with the aid of discontinuous functions. The aim of this project is to establish effective methodologies for the solution of discontinuous phase-change problems. The classic capacitance method, which distributes the effect of any discontinuity present over a finite region (typically an element), can suffer from inaccurate energy transport. Improvement is possible with the application of the classic non-physical enthalpy method. However, this approach is known to suffer with the imposition of material velocity, which gives rise to negative thermal capacitance providing a source of error and instability. In order to improve on the performance of the capacitance method and the classic non-physical enthalpy method, this research introduces a series of new non-physical variables. Firstly, a new non-physical enthalpy is defined via the weak form of the energy transport equation. The classical non-physical enthalpy was defined using a temporal integral term. In the new definition, the non-physical enthalpy involves both a temporal and an advection term, which is shown to avoid the generation of negative capacitance and improve the stability of advection heat transfer in numerical methods. Secondly, control volume analysis is performed on weighted and unweighted forms of the governing energy equation involving non-physical enthalpy. The analysis is shown to reveal non-physical source terms that facilitate the removal of phase-change discontinuities. Thirdly, it is demonstrated in the thesis how a non-physical heat source must be introduced into the governing non-physical transport equation to remove discontinuities arising from non-physical terms related to advection. To demonstrate the accuracy and stability of the new method, it is implemented in the finite element method for both one-dimensional linear rod elements and two dimensional triangular elements. Update techniques and root finding methods, such as the predictor-corrector method, the secant method and the homotopy method, are applied to solve the non-linear system of equations, which are constructed with the new theory. Results returned from the one-dimensional numerical experiments are compared with exact solutions, which show reasonable accuracy. Numerical experiments for isothermal solidification with advection-diffusion in both one and two dimensions demonstrate the feasibility of the new methodology.