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

Mehdi Dehghan (Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran)
Mostafa Abbaszadeh (Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran)
Amirreza Khodadadian (Institute for Analysis and Scientific Computing, Vienna University of Technology (TU Wien), Vienna, Austria and Institute for Applied Mathematics, Leibniz Universität Hannover, Hannover, Germany)
Clemens Heitzinger (Institute for Applied Mathematics, Leibniz Universität Hannover, Hannover, Germany and School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA)

ISSN: 0961-5539

Article publication date: 15 August 2019

Issue publication date: 11 September 2019

172

## Abstract

### Purpose

The 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/approach

At 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.

### Findings

The 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/value

The 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.

## Citation

Dehghan, M., Abbaszadeh, M., Khodadadian, A. and Heitzinger, C. (2019), "Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 8, pp. 2642-2665. https://doi.org/10.1108/HFF-11-2018-0647

## Publisher

:

Emerald Publishing Limited