A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

Zhang, Xiaohua; Zhang, Ping

doi:10.1016/j.amc.2018.07.017

Cited by 12 publications

(8 citation statements)

References 28 publications

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“…Besides, a large numbers of other numerical schemes have been devoted to solving KSE such as backward finite difference scheme proposed by Akrivis and Smyrlis (2004), robust Stackelberg controllability used by Montoya and Breton (2020), new iterative technique applied by Abed (2020), the semi-analytical scheme implemented by Shah, Khan, Baleanu, Kumam, and Arif (2020), discontinuous Galerkin scheme applied by Xu and Shu (2006), tanh-function scheme proposed by Fan (2000), Chebyshev spectral collocation method implemented by Khater and Temsah (2008), compact finite difference method employed by Singh, Arora, and Kumar (2018), compact fourth-order implicit-explicit Runge-Kutta method developed by Bhatt and Chowdhury (2019), compact finite difference method with orthogonal decomposition adopted by Zhang, Zhang, and Ding (2019) and lattice Boltzmann scheme implemented by Lai and Ma (2009). Each method used to solve KSE, still involves certain drawbacks like high arithmetic computations, lower accuracy in terms error, difficulty for computer programming and limitations of certain special cases.…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Jena

Gebremedhin

2021

Arab Journal of Basic and Applied Sciences

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In this article, a nonic B-spline collocation approach is applied to solve the approximate solution of Kuramoto-Sivashinsky equation (KSE). Here the nonlinear term of KSE is linearizing using Taylor series technique. The main objective of this study is to provide a new class of approach by applying some possible higher-order derivatives rather than lower order derivatives of the nonic B-spline on the boundary conditions of KSE in finding the additional constraints, which help us to obtain a unique solution of the problem. The stability analysis is investigated using the Von-Neumann scheme and the proposed scheme is found to be unconditionally stable. The efficiency and accuracy of the proposed approach are demonstrated by two numerical tests and the current results are compared with the exact solution and result of others in the literature.

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Section: Introductionmentioning

confidence: 99%

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Jena

Gebremedhin

2021

Arab Journal of Basic and Applied Sciences

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“…More generally, the general exact solution is hard to obtain and it can just be given in some special forms, therefore, numerical solutions and the corresponding analysis are very important in applications. There have been some numerical methods for solving KdV equation, such as meshless and collocation method [7][8][9][10], Hamiltonian boundary value method [11], Galerkin method [12][13][14][15][16], spectral method [17,18], lattice Boltzmann method [19,20], finite difference method [21][22][23][24][25][26], and others [27][28][29][30]. There are two main types of the problem, one is the periodic boundary value problem and another is the initial-boundary value problem.…”

Section: Introductionmentioning

confidence: 99%

“…Khaled et al [7] considered sinc collocation method to solve the periodic KdV equation and presented several numerical tests to show the accuracy. Zhang et al [26] focused on a high order difference scheme for the same problem. They established a sixth order scheme by proper orthogonal decomposition technique and gave some numerical examples to show the effectiveness.…”

Section: Introductionmentioning

confidence: 99%

A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

Wang

Sun

2020

Numerical Methods Partial

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In this article, we present a two-level implicit difference scheme for Korteweg-de Vires equation with the initial and boundary conditions by the method of order reduction. The truncation error of the difference scheme is analyzed in detail. In the practical computation, the introduced intermediate variable is decoupled in order to reduce the computational cost. It is proved that the difference scheme is solvable by the Browder theorem. The conservation, boundedness, and the unconditional convergence of the numerical solution are also analyzed at length. The convergence order is two both in space and in time in L 2 -norm. The numerical solution is proved to be unique under the optimal step ratio condition. Numerical examples demonstrate that the theoretical analysis is correct.

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“…Chen [5] provided high-order CFDS to solve parabolic equation. Especially, some attempts have been made whose main idea is to combine fourth Runge-Kutta in time and a sixth-order compact finite difference in space (CFDS6) by the researchers [6][7][8]. However, the CFDS6 for parabolic equation, especially the case of desirable accuracy in high dimension, they usually need small spatial discretization or extended finite difference stencils and a small time step which brings about heavy computational loads.…”

Section: Introductionmentioning

confidence: 99%

An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

Zhang

2019

Adv Differ Equ

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In this paper, the combination of efficient sixth-order compact finite difference scheme (E-CFDS6) based proper orthogonal decomposition and Strang splitting method (E-CFDS6-SSM) is constructed for the numerical solution of the multi-dimensional parabolic equation (MDPE). For this purpose, we first develop the CFDS6 to attain a high accuracy for the one-dimensional parabolic equation (ODPE). Then, by the Strang splitting method, we have converted the MDPE into a series of one-dimensional ODPEs successfully, which is easier to implement and program with CFDS6 than an alternating direction implicit method. Finally, we employ proper orthogonal decomposition techniques to improve the computational efficiency of CFDS6 and build the E-CFDS6-SSM with fewer unknowns and sufficiently high accuracy.Six numerical examples are presented to demonstrate that the E-CFDS6-SSM not only can largely alleviate the computational load but also hold a high accuracy and simplify the process of the program for the numerical solution of MDPE.

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A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

Cited by 12 publications

References 28 publications

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

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