The present work focuses on the extension of the Spectral Difference (SD) scheme to the five-equation Baer-Nunziato model for the simulation of immiscible compressible fluids. This five-equation model is augmented with the Allen-Cahn regularisation to avoid both over-diffusion and over-thinning of the phase field representing the interface. In order to preserve contact discontinuities, in the reconstruction step of the SD scheme, a change of variables from conservative to primitive is used. This approach is shown to be beneficial in avoiding pressure oscillations at material interfaces. An extensive series of numerical tests, considering both two- and three-dimensional problems, are performed to assess accuracy and robustness of the present method. Specifically, both laminar and turbulent flows, as well as low-Mach and highly compressible flows, are considered, including cases with and without surface tension. The proposed change of variables is shown to improve the stability of the scheme, significantly reducing pressure oscillations at the material interfaces. This improved robustness enables the method to achieve accurate and stable solutions across a broad range of flow conditions.
