A two-level discretization method for the smagorinsky model

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed and analyzed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We demonstrate numerically that, for an appropriate choice of grids, the two-level algorithm is significantly more efficient than the standard one-level algorithm. We also provide a rigorous numerical analysis of the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (a). Our analysisprovides a mathematical answer to the large eddy simulation mystery of finding the scaling between the filter radius and numerical mesh-size.