Turbulent flows strain resources, both memory and CPU speed. A family of second-order, $G$-stable time-stepping methods proposed by Dahlquist, Liniger, and Nevanlinna (the DLN method) has great accuracy and allows large 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 the DLN method to one corrected Smagorinsky model and provide a detailed numerical analysis of the stability and consistency. We prove that the numerical solutions under arbitrary time step sequences are unconditionally stable in the long term and converge in second order. We also provide error estimates under certain time-step conditions. 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 a flow of energy from unresolved fluctuations to the resolved mean velocity is visible.