Dahlquist, Liniger, and Nevanlinna design a family of one-leg, two-step methods (the DLN method) that is second order, A- and G-stable for arbitrary, non-uniform time steps. Recently, the implementation of the DLN method can be simplified by the refactorization process (adding time filters on backward Euler scheme). Due to these fine properties, the DLN method has strong potential for the numerical simulation of time-dependent fluid models. In the report, we propose a semi-implicit DLN algorithm for the Navier Stokes equations (avoiding non-linear solver at each time step) and prove the unconditional, long-term stability and second-order convergence with the moderate time step restriction. Moreover, the adaptive DLN algorithms by the required error or numerical dissipation criterion are presented to balance the accuracy and computational cost. Numerical tests will be given to support the main conclusions.

翻译：Dahlquist、Liniger和Nevanlinna设计了一族单支两步方法（DLN方法），该方法对于任意非均匀时间步长具有二阶精度、A稳定和G稳定性。近年来，通过重构过程（在向后欧拉格式上添加时间滤波器）简化了DLN方法的实现。基于这些优良特性，DLN方法在含时流体模型的数值模拟中展现出巨大潜力。本文针对Navier Stokes方程提出一种半隐式DLN算法（避免每个时间步求解非线性系统），并在适中的时间步长限制下证明了其无条件长期稳定性与二阶收敛性。此外，基于所需误差或数值耗散判据，我们提出了自适应DLN算法以平衡精度与计算成本。数值实验验证了主要结论。