A nonlinear Galerkin method for the Navier-Stokes equations

doi:10.1016/0045-7825(90)90028-K

Computer Methods in Applied Mechanics and Engineering

Volume 80, Issues 1–3, June 1990, Pages 245-260

https://doi.org/10.1016/0045-7825(90)90028-K Get rights and content

Abstract

Modern large scale computing allows the utilization of a very large number of variables/modes for spatial discretization. Therefore the computer tends to be saturated by computations on small wavelengths that carry a small percentage of the total energy. We advocate the utilization of algorithms treating differently small wavelengths and large wavelengths and we present here an algorithm of this sort, the nonlinear Galerkin method, stemming from the dynamical system theory.

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This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.
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Approximate Inertial Manifolds (AIMs) is approached by multilevel finite element method, which can be referred to as a Post-processed nonlinear Galerkin finite element method, and is applied to the model reduction for fluid dynamics, a typical kind of nonlinear continuous dynamic system from viewpoint of nonlinear dynamics. By this method, each unknown variable, namely, velocity and pressure, is divided into two components, that is the large eddy and small eddy components. The interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially by AIMs, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. As an example, the flow field of incompressible flows around airfoil is simulated numerically and discussed, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less essential degrees-of-freedom which can dominate the dynamic behaviors of the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. In a sense, the small eddy component can be captured by AIMs with fewer grids, and an accurate result can also be obtained.
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A nonlinear Galerkin method for the Navier-Stokes equations

Abstract

J. Funct. Anal.

Physica D

Modelling of the interaction of small and large eddies in two dimensional turbulent flows

Math. Mod. Numer. Anal.

Determining modes and fractal dimension of turbulent flows

J. Fluid Mech.