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

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. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

Researchers developed fully discrete, structure-preserving numerical algorithms of arbitrarily high order for the Allen-Cahn model with a nonlocal constraint, achieving the necessary numerical accuracy and efficiency through energy quadratization methodology. The reformulation of the model into an equivalent one with a quadratic free energy allowed for the preservation of volume conservation and energy dissipative property, while employing distinct temporal discretization methods for fully discrete schemes of arbitrarily higher order.