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Partial differential equations (PDEs) are often dependent on input quantitieswhich are inherently uncertain. To quantify this uncertainty, these PDEs mustbe solved over a large ensemble of parameters. Even for a single realizationthis can a computationally intensive process. In the case of flows governed bythe Navier-Stokes equations, an efficient method has been devised for computingan ensemble of solutions. To further reduce the computational cost of thismethod, an ensemble proper orthogonal decomposition (POD) method was recentlyproposed.  The main contribution of this work is the introduction of POD spatialfiltering for ensemble-POD methods. The POD spatial filter makes possible theconstruction of the Leray ensemble-POD model, which is a regularized reducedorder model for the numerical simulation of convection-dominated flows. TheLeray ensemble-POD model employs the POD spatial filter to smooth (regularize)the convection term in the Navier-Stokes equations and greatly diminishes thenumerical inaccuracies produced by the ensemble-POD method in the numericalsimulation of convection-dominated flows. Specifically, for the numericalsimulation of a convection-dominated two-dimensional flow between two offsetcylinders, we show that the Leray ensemble-POD method yields accurate results,whereas the ensemble-POD is highly inaccurate.  The second contribution of this work is a new numerical discretization of thevariable viscosity ensemble algorithm in which the average viscosity isreplaced with the maximum viscosity. It is shown that this new numericaldiscretization is significantly more stable than those in current use.Furthermore, error estimates for the novel Leray ensemble-POD algorithm withthis new numerical discretization are also proven.

A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows