Error Analysis of the Supplementary Variable Method for the Allen-Cahn Model (Communications in Computational Physics, 2025)
The supplementary variable method has received much attention as a newtechnique to construct structure-preserving algorithm that can preserve the originalenergy law. However, there has been a lack of relevant convergence analysis results,mainly because the method requires solving a nonlinear algebraic equation for the supplementary variable, and the non-uniqueness of its root makes estimating the root’serror very difficult. In this paper, we take the Allen-Cahn model as an example andconstruct a second-order scheme using the supplementary variable method that preserves the original energy dissipation law. We then give a sufficient condition to ensurethat the root of the nonlinear algebraic equation exists uniquely in the neighborhoodof its exact solution. Under this condition, and when the time step is sufficiently small,we establish a rigorous error estimator for this scheme. Finally, we validate the effectiveness of the proposed scheme through several numerical examples.