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Xinyu Cheng, Energy stable semi-implicit schemes for the 2D Allen–Cahn and fractional Cahn–Hilliard equations, IMA Journal of Numerical Analysis, 2025;, draf010, https://doi.org/10.1093/imanum/draf010
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Abstract
In this work, we are interested in a class of numerical schemes for certain phase field models. It is well known that unconditional energy stability (energy decays in time regardless of the size of the time step) provides a fidelity check in practical numerical simulations. In recent work (Li, D. (2022b, Why large time-stepping methods for the Cahn–Hilliard equation is stable. Math. Comp., 91, 2501–2515)), a type of semi-implicit scheme for the Cahn–Hilliard (CH) equation with regular potential was developed satisfying the energy-decay property. In this paper, we extend such semi-implicit schemes to the Allen–Cahn equation and the fractional CH equation with a rigorous proof of similar energy stability. Models in two spatial dimensions are discussed.