Non-linear model reduction for the Navier–Stokes equations using residual DEIM method

• Published in: Journal of computational physics 2014-04, Vol.263, p.1-18
• Main Authors: Xiao, D., Fang, F., Buchan, A.G., Pain, C.C., Navon, I.M., Du, J., Hu, G.
• Format: Article
• Language: English
• Subjects: Computer simulation; Cylinders; Empirical interpolation method; Mathematical analysis; Mathematical models; Navier-Stokes equations; Navier–Stokes; Non-linear model reduction; Nonlinearity; Operators; Petrov–Galerkin; Proper orthogonal decomposition; Reduced order models
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.01.011

This article presents a new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier–Stokes equations. The novelty of the method lies in its treatment of the equation's non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov–Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions. A demonstration of the capabilities of this new approach is provided by solving the incompressible Navier–Stokes equations for simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution's accuracy.