A fast and practical adaptive finite difference method for the conservative Allen–Cahn model in two-phase flow system

We present a simple and practical adaptive finite difference method for the conservative Allen–Cahn–Navier–Stokes system. For the conservative Allen–Cahn equation, we use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. The Navier–Stokes equation is solved in a fully discrete domain with the coarse grid than that for the CAC equation. Various benchmark numerical experiments, such as the pressure jump, droplet deformation in shear flow, falling droplet, and rising bubble, are performed to show that the proposed method is efficient and practical for the simulations of two-phase incompressible flow.