Unconditionally energy-stable time-marching methods for the multi-phase conservative Allen–Cahn fluid models based on a modified SAV approach

Abstract

In this study, we add nonlocal Lagrange multipliers to the Allen–Cahn (AC) equation to establish the N-component conservative Allen–Cahn (CAC) system and the ternary conservative Allen–Cahn–Navier–Stokes (CACNS) system which conserve the mass for each phase. Then, we construct linear, totally decoupled, and energy dissipative methods for those multi-component phase-field. In the classical scalar auxiliary variable (SAV) approach, the phase variables and the scalar auxiliary variables are coupled, which need extra computational time to decouple. In this paper, we adopt a variant of SAV approach in which all variables are totally decoupled and can be calculated in a step-by-step manner. The linear multigrid algorithm is adopted to accelerate convergence. We further analytically prove the unique solvability, modified energy dissipation law, and mass conservation in time-discretized version. We demonstrate the performance of the proposed scheme via various two-dimensional (2D) and three-dimensional (3D) numerical simulations. Contrast tests between the N-component CAC model and the N-component Cahn–Hilliard (CH) model show that the N-component CAC model is efficient.