Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

• Published in: Applied numerical mathematics Vol. 170; pp. 321 - 339
• Main Authors: Hong, Qi, Gong, Yuezheng, Zhao, Jia, Wang, Qi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2021
• Subjects: Cosine pseudo-spectral method; Energy stable schemes; Linear high-order schemes; Nonlocal Allen-Cahn model; Symplectic Runge-Kutta method; Nonlocal Allen-Cahn model; Linear high-order schemes; Cosine pseudo-spectral method; Symplectic Runge-Kutta method; Energy stable schemes
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2021.08.002

We develop fully discrete structure-preserving numerical algorithms of arbitrarily high order for the Allen-Cahn model with a nonlocal constraint subject to the Neumann boundary condition. Using the energy quadratization methodology, we reformulate the thermodynamically consistent model into an equivalent one with a quadratic free energy. For the reformulated model, we first apply a Cosine pseudo-spectral approximation in space to arrive at a semi-discrete system that inherits the volume conservation and energy dissipative property; then we use two distinct temporal discretization methods to derive fully discrete schemes of arbitrarily higher order. One is based on the symplectic Runge-Kutta (RK) method and the other is a linearized Runge-Kutta method by the prediction-correction strategy. The fully discrete schemes preserve both volume and the energy dissipative property. In addition, we show that the liner system resulting from the schemes warrants the unique solvability. A fast solver combined with the discrete Cosine transform (DCT) is exploited to implement the high-order scheme efficiently. Extensive numerical examples are presented to show the efficiency and accuracy of the newly proposed methods.