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

Dehghan, Mehdi; Abbaszadeh, Mostafa; Khodadadian, Amirreza; Heitzinger, Clemens

doi:10.1108/hff-11-2018-0647

Cited by 19 publications

(4 citation statements)

References 79 publications

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“…Dehghan and Mohammadi (2020) used the boundary knot method for the solution of twodimensional advection reaction-diffusion and Brusselator equations. The main aim of Dehghan et al (2019) is to develop a reduced order discontinuous Galerkin method for solving the generalized Swift-Hohenberg equation with application in biological science and mechanical engineering. A numerical solution of a family of generalized fifth-order Korteweg-de Vries equations using the RBF MOL has been presented.…”

Section: Application Of Spd-rbf Methods Of Linesmentioning

confidence: 99%

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Mesgarani

Kermani

Abbaszadeh

2021

HFF

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Purpose The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients. Design/methodology/approach The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well. Findings Several test problems are provided to confirm the validity and efficiently of the proposed method. Originality/value For the first time, some famous examples are solved by using the proposed high-order technique.

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Section: Application Of Spd-rbf Methods Of Linesmentioning

confidence: 99%

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Mesgarani

Kermani

Abbaszadeh

2021

HFF

Self Cite

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“…Authors of Dehghan and Mohammadi (2020) used the boundary knot method for the solution of two-dimensional (2D) advection reaction-diffusion and Brusselator equations. Also, the main aim of Dehghan et al (2019) is to develop a reduced-order discontinuous Galerkin method for solving the generalized Swift-Hohenberg equation with application in biological science and mechanical engineering. A numerical solution of a family of generalized fifth-order Korteweg-de Vries equations using the RBF method of lines has been presented by Mohyud-Din et al (2012).…”

Section: Local Radial Basis Function-differential Quadrature Methodsmentioning

confidence: 99%

The local meshless collocation method for solving 2D fractional Klein-Kramers dynamics equation on irregular domains

Abbaszadeh

Pourbashash

Oshagh

2021

HFF

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Purpose This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains. Design/methodology/approach The main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space. Findings First, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders O(τ1+γ) and O(τ2−γ) for 0 < γ < 1. Finally, some numerical examples have been presented to show the high accuracy and acceptable results of the proposed technique. Originality/value The proposed numerical technique is flexible for different computational domains.

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“…Liu [16] considered two linear, second-order and unconditionally energy stable schemes by linear invariant energy quadratization and modified scalar auxiliary variable approaches. Dehghan et al [17] combined the proper orthogonal decomposition approach and the local discontinuous Galerkin technique, and discussed the energy stability. Sun et al [21] proposed an adaptive BDF2 scheme for the Swift-Hohenberg equation, and proved the energy stability and the convergence.…”

Section: Introductionmentioning

confidence: 99%

Energy stable and $L^2$ norm convergent BDF3 scheme for the Swift-Hohenberg equation

Zhang¹,

Xue²,

Sun³

2023

Preprint

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A fully discrete implicit scheme is proposed for the Swift-Hohenberg model, combining the third-order backward differentiation formula (BDF3) for the time discretization and the second-order finite difference scheme for the space discretization. Applying the Brouwer fixed-point theorem and the positive definiteness of the convolution coefficients of BDF3, the presented numerical algorithm is proved to be uniquely solvable and unconditionally energy stable, further, the numerical solution is shown to be bounded in the maximum norm. The proposed scheme is rigorously proved to be convergent in L 2 norm by the discrete orthogonal convolution (DOC) kernel, which transfer the four-level-solution form into the three-levelgradient form for the approximation of the temporal derivative. Consequently, the error estimate for the numerical solution is established by utilization of the discrete Gronwall inequality. Numerical examples in 2D and 3D cases are provided to support the theoretical results.

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

Cited by 19 publications

References 79 publications

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

The local meshless collocation method for solving 2D fractional Klein-Kramers dynamics equation on irregular domains

Energy stable and $L^2$ norm convergent BDF3 scheme for the Swift-Hohenberg equation

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