The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model
<|reference_start|>The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model: We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional 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.<|reference_end|>
@article{chen2024the,
    title={The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn
  Model},
    author={YiMing Chen, Dianlun Luo, Wenlong Pei, and Yulong Xing},
    journal={arXiv preprint arXiv:2409.19481},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.19481},
    primaryClass={math.NA cs.NA}
}