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.