Efficient implementation of the Allen–Cahn phase-field method for material interface tracking

We consider a modified version of the conservative Allen–Cahn phase-field method for tracking material interfaces. By removing the diffusive term and discretizing the advective terms with a first-order upwind scheme, we achieve a substantial reduction in numerical diffusion, resulting in sharper interfaces for a given compression parameter. The resulting scheme is monotone, leading to key numerical properties such as boundedness and convergence to weak entropy solutions. Notably, the new method permits a significantly larger time step compared to standard implementations, especially in multiple spatial dimensions. Numerical tests support these theoretical findings and demonstrate effectiveness and robustness of the new algorithm across a range of multiphase flow problems.