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A Two-Level Discretization Method for the Smagorinsky Model

2008; Society for Industrial and Applied Mathematics; Volume: 7; Issue: 2 Linguagem: Inglês

10.1137/070704812

1540-3467

Jeff Borggaard, Traian Iliescu, Hyesuk Lee, John Paul Roop, Hyunjin Son,

Meteorological Phenomena and Simulations

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 ($\delta$). Our analysis provides a mathematical answer to the large eddy simulation mystery of finding the scaling between the filter radius and numerical mesh-size.