TY - JOUR

T1 - Equation-free/Galerkin-free POD-assisted computation of incompressible flows

AU - Sirisup, Sirod

AU - Karniadakis, George Em

AU - Xiu, Dongbin

AU - Kevrekidis, Ioannis G.

N1 - Funding Information:
The first author gratefully acknowledges the DPST (Development and Promotion of Science and Technology Talents) project from Thailand for providing his scholarship during his graduate studies at Brown University. This work was supported by ONR and NSF, and computations were performed at the facilities of NCSA (University of Illinois at Urbana-Champaign) and at DoDs NAVO MSRC.

PY - 2005/8/10

Y1 - 2005/8/10

N2 - We present a Galerkin-free, proper orthogonal decomposition (POD)-assisted computational methodology for numerical simulations of the long-term dynamics of the incompressible Navier-Stokes equations. The approach is based on the "equation-free" framework: we use short, appropriate initialized bursts of full direct numerical simulations (DNS) of the Navier-Stokes equations to observe, estimate, and accelerate, through "projective integration", the evolution of the flow dynamics. The main assumption is that the long-term dynamics of the flow lie on a low-dimensional, attracting, and invariant manifold, which can be parametrized, not necessarily spanned, by a few POD basis functions. We start with a discussion of the consistency and accuracy of the approach, and then illustrate it through numerical examples: two-dimensional periodic and quasi-periodic flows past a circular cylinder. We demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long-term asymptotic state of the flow in cases where traditional Galerkin-POD models fail. The approach trades the overhead involved in developing POD-Galerkin and POD-nonlinear Galerkin codes, for the repeated (yet short, and on demand) use of an existing full DNS simulator. Moreover, since in this approach the POD modes are used to observe rather than span the true system dynamics, the computation is much less sensitive than POD-Galerkin to values of the system parameters (e.g., the Reynolds number) and the particular simulation data ensemble used to obtain the POD basis functions.

AB - We present a Galerkin-free, proper orthogonal decomposition (POD)-assisted computational methodology for numerical simulations of the long-term dynamics of the incompressible Navier-Stokes equations. The approach is based on the "equation-free" framework: we use short, appropriate initialized bursts of full direct numerical simulations (DNS) of the Navier-Stokes equations to observe, estimate, and accelerate, through "projective integration", the evolution of the flow dynamics. The main assumption is that the long-term dynamics of the flow lie on a low-dimensional, attracting, and invariant manifold, which can be parametrized, not necessarily spanned, by a few POD basis functions. We start with a discussion of the consistency and accuracy of the approach, and then illustrate it through numerical examples: two-dimensional periodic and quasi-periodic flows past a circular cylinder. We demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long-term asymptotic state of the flow in cases where traditional Galerkin-POD models fail. The approach trades the overhead involved in developing POD-Galerkin and POD-nonlinear Galerkin codes, for the repeated (yet short, and on demand) use of an existing full DNS simulator. Moreover, since in this approach the POD modes are used to observe rather than span the true system dynamics, the computation is much less sensitive than POD-Galerkin to values of the system parameters (e.g., the Reynolds number) and the particular simulation data ensemble used to obtain the POD basis functions.

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U2 - 10.1016/j.jcp.2005.01.024

DO - 10.1016/j.jcp.2005.01.024

M3 - Article

AN - SCOPUS:18844459925

SN - 0021-9991

VL - 207

SP - 568

EP - 587

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 2

ER -