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Research Papers: Techniques and Procedures

Equation-Free/Galerkin-Free Reduced-Order Modeling of the Shallow Water Equations Based on Proper Orthogonal Decomposition

[+] Author and Article Information
Vahid Esfahanian

Department of Mechanical Engineering, University of Tehran, North Kargar Avenue, Tehran 11365-4565, Iranevahid@ut.ac.ir

Khosro Ashrafi1

Faculty of Environment, University of Tehran, Enghelab Avenue, Tehran 14155-6135, Irankhashrafi@ut.ac.ir

1

Corresponding author.

J. Fluids Eng 131(7), 071401 (Jun 23, 2009) (13 pages) doi:10.1115/1.3153368 History: Received March 01, 2008; Revised April 16, 2009; Published June 23, 2009

In this paper, two categories of reduced-order modeling (ROM) of the shallow water equations (SWEs) based on the proper orthogonal decomposition (POD) are presented. First, the traditional Galerkin-projection POD/ROM is applied to the one-dimensional (1D) SWEs. The result indicates that although the Galerkin-projection POD/ROM is suitable for describing the physical properties of flows (during the POD basis functions’ construction time), it cannot predict that the dynamics of the shallow water flows properly as it was expected, especially with complex initial conditions. Then, the study is extended to applying the equation-free/Galerkin-free POD/ROM to both 1D and 2D SWEs. In the equation-free/Galerkin-free framework, the numerical simulation switches between a fine-scale model, which provides data for construction of the POD basis functions, and a coarse-scale model, which is designed for the coarse-grained computational study of complex, multiscale problems like SWEs. In the present work, the Beam & Warming and semi-implicit time integration schemes are applied to the 1D and 2D SWEs, respectively, as fine-scale models and the coefficients of a few POD basis functions (reduced-order model) are considered as a coarse-scale model. Projective integration is applied to the coarse-scale model in an equation-free framework with a time step grater than the one used for a fine-scale model. It is demonstrated that equation-free/Galerkin-free POD/ROM can resolve the dynamics of the complex shallow water flows. Moreover, the computational cost of the approach is less than the one for a fine-scale model.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Three different initial conditions for the POD/ROM of the 1D SWEs: (a) smooth initial condition, (b) high frequency initial condition, and (c) hydraulic jump initial condition

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Figure 2

Eigenvalues of the correlation matrix C with 0.5 s time interval for snapshots: (a) flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition

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Figure 3

Time evolutions of the first five aks: (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition

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Figure 4

A 200 s predicted height field by the Galerkin-projection POD/ROM method compared to the Beam & Warming method, the POD basis functions’ construction time is 50 s and the numbers of modes are 10, 44, and 79 modes for (a), (b), and (c), respectively. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 5

A 400 s predicted height field by the Galerkin-projection POD/ROM method compared to the Beam & Warming method, the POD basis functions’ construction time is 50 s, and the numbers of modes are 10, 44, and 79 modes for (a), (b), and (c), respectively. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 6

A 200 s predicted velocity field by the Galerkin-projection POD/ROM method compared to the Beam & Warming method, the POD basis functions’ construction time is 50 s, and the numbers of modes are 10, 44, and 79 modes for (a), (b), and (c), respectively. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 7

A 400 s predicted velocity field by the Galerkin-projection POD/ROM method compared to the Beam & Warming method, the POD basis functions’ construction time is 50 s, and the numbers of modes are 10, 44, and 79 modes for (a), (b), and (c), respectively. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 8

Sketch of a projective integration over one global time step (13)

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Figure 9

A 200 s predicted height field by the equation-free/Galerkin-free POD/ROM method compared to the Beam & Warming method. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 10

A 400 s predicted height field by the equation-free/Galerkin-free POD/ROM method compared to the Beam & Warming method. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 11

A 200 s predicted velocity field by the equation-free/Galerkin-free POD/ROM method compared to the Beam & Warming method. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 12

A 400 s predicted velocity field by the equation-free/Galerkin-free POD/ROM method compared to the Beam & Warming method. (a) Flow with a smooth initial condition, (b) flow with a high frequency initial condition, and (c) flow with a hydraulic jump initial condition.

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Figure 13

A 200 s predicted velocity field by the equation-free/Galerkin-free POD/ROM method compared to Beam & Warming method in fine and coarser grids (the CPU-time of the coarser grid is equal to POD/ROM)

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Figure 14

Initial condition for potential vorticity

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Figure 15

Potential vorticity patterns after 4 days of integration (top), 8 days of integration (middle), and 12 days of integration (bottom). (a) 65×65 grid points, (b) 129×129 grid points, and (c) 257×257 grid points.

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Figure 16

Comparison of potential vorticity patterns between semi-implicit time integration (fine-scale model) and the equation-free/Galerkin-free POD/ROM of the SWEs after 4 days of integration (top), 8 days of integration (middle), and 12 days of integration (bottom). (a) Semi-implicit time integration and (b) equation-free/Galerkin-free POD/ROM.

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