A class of high-order compact difference schemes for solving the Burgers’ equations

Yang, Xiaoxia; Ge, Yongbin; Zhang, Lin

doi:10.1016/j.amc.2019.04.023

Cited by 25 publications

(21 citation statements)

References 11 publications

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“…We find that the scheme in Ref. [42] is fourth order in space, and the schemes in Ref. [44] are fourth and sixth order, respectively.…”

Section: Numerical Experimentsmentioning

confidence: 68%

“…[44] and the scheme in Ref. [42] all achieve theoretical accuracy. At the same time, the sixth-order scheme in Ref.…”

Section: Numerical Experimentsmentioning

confidence: 79%

“…We will compare the numerical results computed by the present HOC schemes with those derived by the methods in the existing literature. We note that for the methods in [42,43], we use PHOEBESolver software (http://www.phoebesolver. com/webpde/main/index) (accessed on 2 Feburary 2022) to calculate.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

Jian-ying

Wang

2022

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In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.

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“…We find that the scheme in Ref. [42] is fourth order in space, and the schemes in Ref. [44] are fourth and sixth order, respectively.…”

Section: Numerical Experimentsmentioning

confidence: 68%

“…[44] and the scheme in Ref. [42] all achieve theoretical accuracy. At the same time, the sixth-order scheme in Ref.…”

Section: Numerical Experimentsmentioning

confidence: 79%

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High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

Jian-ying

Wang

2022

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“…High order compact difference schemes can increase the accuracy of standard difference approximations by using compact stencils. The fourth-order compact finite difference methods were proposed for second order elliptic/parabolic problems [18][19][20][21][22][23][24][25][26][27][28][29]. The attractive merit of such a method is that although a similar compact stencil as the standard second-order central finite difference method is used, it still achieves fourth-order accuracy.…”

Section: Introductionmentioning

confidence: 99%

Compact difference scheme for two‐dimensional fourth‐order nonlinear hyperbolic equation

Yang

Chen

2020

Numerical Methods Partial

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High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.

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“…Numerical solutions of this equation, used in modelling the solitary wave and travelling waves, have been studied by the researchers due to the limitations in analytical solutions. Numerical solutions, which have been investigated recently, have been obtained by least squares, splitting, homotopy perturbation, finite difference and quadrature methods [3][4][5][6][7][8][9][10]. In parallel with these studies, we used the quintic B-spline collocation method for space and the various-order single step methods for time discretization that were not implemented before.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Accurate Numerical Algorithm for the Numerical Solution of Burgers’ Equation

Görgülü

2019

Muş Alparslan Üniversitesi Fen Bilimleri Dergisi

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In this study, a high accurate numerical solution of the Burgers' equation is obtained. For this, the collocation method based on quintic B-spline functions for space discretization and the fourth-order single step method for time discretization are used. In order to see the efficiency of the algorithm, a test problem with an analytical solution is discussed and compared with the numerical solution obtained.

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A class of high-order compact difference schemes for solving the Burgers’ equations

Cited by 25 publications

References 11 publications

High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

Compact difference scheme for two‐dimensional fourth‐order nonlinear hyperbolic equation

A High-Order Accurate Numerical Algorithm for the Numerical Solution of Burgers’ Equation

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