  A diffused-interface approach based on the Allen-Cahn phase field equation is
developed within a high-order Discontinuous Galerkin framework. The interface
capturing technique is based on the balance between explicit diffusion and
sharpening terms in the phase field equation, where the former term involves
the computation of the local interface normal vectors. Due to the well-known
Gibbs phenomenon encountered in high-order discretisations of steep profiles
such as shocks and/or interfaces, the accurate evaluation of the normal vector
requires special consideration. To this end, a non-linear preconditioning
strategy is proposed in this work where an additional smooth level-set function
advected by the velocity field is used for the evaluation of the normal
vectors. It is shown that for appropriate choices of numerical fluxes and
parameters of the model, the phase field remains bounded without any need for
explicit regularisation. The proposed diffused-interface technique is
implemented within a five equation model for fully compressible two-phase
flows. In order to preserve isolated interfaces, a quasi-conservative
discretisation of the five equation model is employed. A series of numerical
experiments of increasing complexity are performed in order to assess the
accuracy and robustness of the developed methodology, including two-phase flows
involving viscous effects, gravitational forces, and surface tension.
