Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn–Hilliard and Allen–Cahn equations

Abstract

Comparing with the classic phase filed models, the fractional models such as time fractional Allen-Cahn and Cahn-Hilliard equations are equipped with Caputo fractional derivative and can describe more practical phenomena for modeling phase transitions. In this paper, we construct two accurate and efficient linear algorithms for the time fractional Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potential. The main contribution is that we have proved the unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn models and their semi-discrete schemes carefully and rigorously. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.

An efficient linearly implicit and energy‐conservative scheme for two dimensional Klein–Gordon–Schrödinger equations

“…Based on the idea of the Lagrange multiplier method, the original problem transforms to an equivalent system with quadratic energy form by introducing some arbitrary variables, and the IEQ approach aims to develop the linear, implicit and unconditionally energy stable scheme. Thanks to the advantages, the IEQ approach has been successfully applied to solve gradient flows, such as phase field elastic bending energy models [29,48], Allen-Cahn and Cahn-Hilliard equations [6,35], Peng-Robinson equation of state [23,36,51]. Shen et al [37] implemented the idea of IEQ approach using a scalar auxiliary variable and called it the SAV method to circumvent the disadvantages of the IEQ approach.…”