Navier-Stokes方程的半隐式DLN算法
The Semi-implicit DLN Algorithm for the Navier Stokes Equations
AI总结：
针对Navier-Stokes方程提出半隐式DLN算法，避免逐时间步非线性求解，证明其长期稳定性与二阶收敛性，并给出误差或耗散驱动的自适应版本。
摘要：
Dahlquist、Liniger和Nevanlinna设计了一族单腿两步方法（DLN方法），该方法对任意非均匀时间步均为二阶且A-稳定和G-稳定。最近，DLN方法的实现可通过重构过程（在向后Euler格式上添加时间滤波器）得到简化。由于这些优良性质，DLN方法在非定常流体模型的数值模拟方面具有很大潜力。在本报告中，我们提出了一种用于Navier-Stokes方程的半隐式DLN算法（避免在每个时间步使用非线性求解器），并在适度的时间步限制下证明了其无条件长期稳定性和二阶收敛性。此外，为平衡精度与计算成本，我们还提出了基于所需误差或数值耗散准则的自适应DLN算法。数值测试将用于支持主要结论。
英文摘要：
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.
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