We propose a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i) In the first step, we use explicit ROM spatial filtering of the nonlinear PDE to construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. We emphasize that the new DDF-ROM is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments.We investigate the DDF-ROM in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds number Re = 100. The DDF-ROM is significantly more accurate than the standard projection ROM. Furthermore, the computational costs of the DDF-ROM and the standard projection ROM are similar, both costs being orders of magnitude lower than the computational cost of the full order model. We also compare the new DDF-ROM with modern ROM closure models in the numerical simulation of the 1D Burgers equation. The DDF-ROM is more accurate and significantly more efficient than these ROM closure models.where a is the vector of unknown ROM coefficients and A ∈ R r×r , B ∈ R r×r×r are ROM operators. (vi) In an offline stage, compute the ROM operators. (vii) In an online stage, repeatedly use the Proj-ROM (1.1) (for various parameter settings

Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

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An efficient algorithm is proposed and studied for computing flow ensembles of incompressible magnetohydrodynamic (MHD) flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through a clever algorithm (adapted from a breakthrough idea of Jiang and Layton [23]) that, at each time step, uses the same matrix for each of the J systems solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Elsässer variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments.

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We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier-Stokes equations (NSE). The method combines a linear treatment of the advection term, together with a stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O(∆t 2) accurate. The idea of 'curvature stabilization' is new to CFD and is intended as an improvement over the commonly used 'speed stabilization', which is only first order accurate in time and can have an adverse affect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows.

For reduced order models (ROMs) of fluid flows, we investigate theoretically and computationally whether differentiation and ROM spatial filtering commute, i.e., whether the commutation error (CE) is nonzero. We study the CE for the Laplacian and two ROM filters: the ROM projection and the ROM differential filter. Furthermore, when the CE is nonzero, we investigate whether it has any significant effect on ROMs that are constructed by using spatial filtering. As numerical tests, we use the Burgers equation with viscosities ν = 10 −1 and ν = 10 −3 and a 2D flow past a circular cylinder at Reynolds numbers Re = 1 and Re = 100. Our investigation shows that: (i) the CE exists; and (ii) the CE has a significant effect on ROM development for low Reynolds numbers, but not so much for higher Reynolds numbers.

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