(untitled) (Communications in Computational Physics, 2025)
The supplementary variable method has received much attention as a new technique to construct structure-preserving algorithm that can preserve the original energy 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’s error very difficult. In this paper, we take the Allen-Cahn model as an example and construct a second-order scheme using the supplementary variable method that preserves the original energy dissipation law. We then give a sufficient condition to ensure that the root of the nonlinear algebraic equation exists uniquely in the neighborhood of 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.