Genuinely nonlinear models for convection-dominated problems

AbstractThis paper introduces a general, nonlinear subgrid-scale (SGS) model, having boundedartificial viscosity, for the numerical simulation of convection-dominated problems. We also present a numerical comparison (error analysis and numerical experiments) between this model and the most common SGS model of Smagorinsky, which uses a p-Laplacian regularization. The numerical experiments for the 2-D convection-dominated convection-diffusion test problem show a clear improvement in solution quality for the new SGS model. This improvement is consistent with the bounded amount of artificial viscosity introduced by the new SGS model in the sharp transition regions

Variable Time Step Method of DAHLQUIST, LINIGER and NEVANLINNA (DLN) for a Corrected Smagorinsky Model

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