Turbulent flows strain resources, both memory and CPU speed. The DLN method has greater accuracy and allows larger time steps, requiring less memory and fewer FLOPS. The DLN method can also be implemented adaptively. The classical Smagorinsky model, as an effective way to approximate a (resolved) mean velocity, has recently been corrected to represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. In this paper, we apply a family of second-order, G-stable time-stepping methods proposed by Dahlquist, Liniger, and Nevanlinna (the DLN method) to one corrected Smagorinsky model and provide the detailed numerical analysis of the stability and consistency. We prove that the numerical solutions under any arbitrary time step sequences are unconditionally stable in the long term and converge at second order. We also provide error estimate under certain time step condition. Numerical tests are given to confirm the rate of convergence and also to show that the adaptive DLN algorithm helps to control numerical dissipation so that backscatter is visible.

翻译：湍流流动会大量消耗计算资源（包括内存与CPU速度）。DLN方法具有更高精度，允许采用更大的时间步长，从而减少内存需求与浮点运算次数。该方法还可实现自适应计算。经典Smagorinsky模型作为近似（已解析）平均速度的有效途径，近期已被修正以表征能量从未解析脉动向（已解析）平均速度的传递。本文将Dahlquist、Liniger与Nevanlinna提出的二阶G-稳定时间步长算法族（DLN方法）应用于一个修正Smagorinsky模型，并对其进行详细的稳定性与相容性数值分析。我们证明：在任意时间步长序列下，数值解长期无条件稳定且具有二阶收敛性。同时给出了特定时间步长条件下的误差估计。数值实验验证了收敛阶，并表明自适应DLN算法有助于控制数值耗散，从而能够呈现反向散射现象。