A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

Zhang, Ping; Zhang, Xiaohua; Xiang, Hui; Song, Laizhong

doi:10.1080/10407782.2016.1177327

Cited by 23 publications

(7 citation statements)

References 29 publications

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“…The POD technique can be found in several research papers for solving different physical models The POD technique is considered by many scholars (Chaturantabut, 2009; Chaturantabut and Sorensen, 2012; Fang et al , 2009; Lin et al , 2017; Ravindran, 2000a; Ravindran, 2000b; Ştefănescu and Navon, 2013; Ştefănescu et al , 2014; Xiao et al , 2015b). The POD approach has been used to solve the multi-species host-parasitoid system (Dimitriu et al , 2015), compressible fluid and fractured solid (Fang et al , 2009; Ravindran, 2000a; Ravindran, 2000b; Xiao et al , 2017), Pacific Ocean model (Cao et al , 2007), shallow water model (Ştefănescu and Navon, 2013; Ştefănescu et al , 2014), 2D Burgers equation (Wang et al , 2016), multiphase porous media flows (Xiao et al , 2015b), Navier–Stokes equations (Xiao et al , 2014), fluid-structure interaction (FSI) (Xiao et al , 2013), dynamic PDEs based on the Smolyak sparse grid collocation (Xiao et al , 2015a), transient heat conduction problems (Zhang and Xiang, 2015), convection-diffusion problems (Zhang et al , 2016) and incompressible Navier–Stokes equation (Dehghan and Abbaszadeh, 2016a; Du et al , 2012; Du et al , 2013; Luo et al , 2008; Xiao et al , 2015b).…”

Section: Proper Orthogonal Decomposition (Pod) Methodsmentioning

confidence: 99%

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

Dehghan

Abbaszadeh

Khodadadian

et al. 2019

HFF

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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.

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Section: Proper Orthogonal Decomposition (Pod) Methodsmentioning

confidence: 99%

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

Dehghan

Abbaszadeh

Khodadadian

et al. 2019

HFF

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show abstract

“…Due to the advantages of the meshless method, some meshless methods have been developed for the CDR problems [48,49]. Zhang and Xiang discussed the numerical solutions of the CDR equation with small diffusion by the VMEFG method [50,51]. Li et al also applied the local Petrov Galerkin method to the multi-dimensional CDR equations based on the radial basis function [52].…”

Section: Introductionmentioning

confidence: 99%

A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems

et al. 2021

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By introducing the dimension splitting method (DSM) into the generalized element-free Galerkin (GEFG) method, a dimension splitting generalized interpolating element-free Galerkin (DS-GIEFG) method is presented for analyzing the numerical solutions of the singularly perturbed steady convection–diffusion–reaction (CDR) problems. In the DS-GIEFG method, the DSM is used to divide the two-dimensional CDR problem into a series of lower-dimensional problems. The GEFG and the improved interpolated moving least squares (IIMLS) methods are used to obtain the discrete equations on the subdivision plane. Finally, the IIMLS method is applied to assemble the discrete equations of the entire problem. Some examples are solved to verify the effectiveness of the DS-GIEFG method. The numerical results show that the numerical solution converges to the analytical solution with the decrease in node spacing, and the DS-GIEFG method has high computational efficiency and accuracy.

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“…A new nonintrusive model reduction method is proposed in Reference for the Navier‐Stokes equations based on the radial basis function (RBF) multidimensional interpolation instead of the traditional approach of projecting the equations onto the reduced space. A fast and stabilized meshless method that combines variational multiscale element‐free Galerkin (VMEFG) method and POD method is developed in Reference to solve convection‐diffusion problems. The POD technique is applied for the meshless method in Reference for transient heat conduction problems.…”

Section: Introductionmentioning

confidence: 99%

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

Abbaszadeh

Dehghan

Navon

2020

Numerical Methods in Fluids

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Summary In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element‐free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced‐order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced‐order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two‐dimensional cases.

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A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

Cited by 23 publications

References 29 publications

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 Swift-Hohenberg equation

A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

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