On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local–nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local–nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It is found that the BBLM formulation can better maintain small features, but the RSLM formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSLM formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.