Journal ArticleEfficient Variable Time-stepping Adaptive DLN Algorithms for the Allen-Cahn EquationChen, Yiming; Luo, Dianlun; Pei, Wenlong; Xing, Yulong<title>Abstract</title> <p>We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditionally non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time-adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time-adaptive DLN methods.</p>Springer2025-08-0110633215Journal of Scientific Computing10420885-7474https://doi.org/10.1007/s10915-025-02980-42309590National Science Foundation