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, the accurate evaluation of the normal vectors requires special consideration. To this end, a non-linear preconditioning strategy is proposed 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. 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.
