This report considers a variable time-stepping algorithm proposed by Dahlquist, Liniger and Nevanlinna and discusses its application to the unsteady Stokes/Darcy model. Although long-time forgotten and little explored, the algorithm performs advantages in variable time-stepping analysis of various fluid flow systems, including the coupled Stokes/Darcy model. We first prove that the approximate solutions to the unsteady Stokes/Darcy model are unconditionally stable due to the $G$-stability of the method. Then variable time-stepping error analysis follows from the combination of $G$-stability and consistency of the method. Numerical experiments verify the theoretical results, demonstrating both accuracy and stability of the algorithm.

本报告考虑了Dahlquist，Liniger和Nevanlinna提出的可变时步算法，并讨论了其在非定常Stokes / Darcy模型中的应用。尽管长期被遗忘并且很少被探索，但是该算法在各种流体流系统的可变时间步长分析中具有优势，包括耦合的Stokes / Darcy模型。我们首先证明了非定常Stokes / Darcy模型的近似解是无条件稳定的，这是由于$G$-方法的稳定性。然后结合以下方法进行可变的时间步长误差分析$G$-方法的稳定性和一致性。数值实验验证了理论结果，证明了算法的准确性和稳定性。