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Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized... PurposeThe current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.Design/methodology/approachAt first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.FindingsThe results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.Originality/valueThe local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Numerical Methods for Heat & Fluid Flow Emerald Publishing

Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0961-5539
DOI
10.1108/HFF-11-2018-0647
Publisher site
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Abstract

PurposeThe current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.Design/methodology/approachAt first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.FindingsThe results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.Originality/valueThe local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

Journal

International Journal of Numerical Methods for Heat & Fluid Flow – Emerald Publishing

Published: Aug 5, 2019

References

  • Numerical simulation of thermal wave propagation during laser processing of thin films
  • Analytical and numerical results for the Swift–Hohenberg equation
  • Unified analysis of discontinuous Galerkin methods for elliptic problems
  • A real distinct poles exponential time differencing scheme for reaction-diffusion systems
  • On discrete maximum principles for discontinuous Galerkin methods
  • The proper orthogonal decomposition in the analysis of turbulent flows
  • A POD reduced order model for resolving angular direction in neutron/photon transport problems
  • A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition