We have developed a full-explicit scheme for solving low Mach number two-phase flows based on weakly compressible assumption with diffuse-interface method of phase-field equation to avoid solving pressure Poisson equation. Compressible Navier-Stokes equa- tions are divided into the Euler equation and other terms; viscosity, surface tension and gravity terms. For computational efficiency and the reduction of implementation difficulty of AMR method, the directional-splitting method is introduced to apply the characteristic method which a semi-Lagrangian scheme is applicable to. We have also implemented a tree-based AMR method to adapt fine meshes near the gas-liquid interface by using a level-set function reconstructed from the phase-field variable.
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