基于广义梯形公式的二维Burgers方程三阶数值解法

刘陶; 刘利斌

doi:10.21656/1000-0887.470066

基于广义梯形公式的二维Burgers方程三阶数值解法

doi: 10.21656/1000-0887.470066

刘陶,
刘利斌^,

南宁师范大学广西应用数学中心, 南宁 530100

基金项目:

国家自然科学基金(12361087)

广西自然科学基金(2025GXNSFAA069738)

详细信息

作者简介:
刘陶(1999—)，男,硕士生(E-mail: taoledu@163.com);刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

通讯作者:
刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

中图分类号: O241.82
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- 文章访问数: 118
- HTML全文浏览量: 26
- PDF下载量: 15
- 被引次数: 0
出版历程
- 收稿日期: 2026-03-17
- 修回日期: 2026-03-17
- 网络出版日期: 2026-09-24
- 刊出日期: 2026-09-01

A 3rd-Order Numerical Method for 2D Burgers Equations Based on the Generalized Trapezoid Formula

Liu Tao,
Liu Libin^,

Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, P.R. China

Funds:

The National Science Foundation of China(12361087)

摘要

摘要: Burgers方程因其对非线性传输与耗散机制的精准描述,在流体力学等多个领域有着广泛的应用.本文研究了一类二维非线性Burgers方程的数值解法.首先，通过Hopf-Cole变换将Burgers方程转换成一类热传导方程,在空间方向利用高阶重心有理谱逼近,得到半离散格式.然后，在时间方向采用含参数δ的广义梯形公式离散,从而得到变换后方程的全离散格式.当参数δ=1/3时,时间方向截断误差达到三阶精度.最后，借助数值算例验证了截断误差分析的正确性.
- Burgers方程 /
- HopfCole /
- 高阶重心有理插值配置法 /
- 广义梯形公式
Abstract: The Burgers equation has been widely applied in many fields such as fluid mechanics due to its accurate description of nonlinear transport and dissipation mechanisms. A numerical method for a class of 2D nonlinear Burgers equations was studied. Firstly, the Burgers equation was transformed into a class of heat conduction equations via the Hopf-Cole transformation. Then, the high-order barycentric rational spectral approximation was used in the spatial direction to obtain the semi-discrete scheme, and the generalized trapezoidal formula with parameter δ was adopted in the temporal direction, thus to obtain the fully discrete scheme for the transformed equation. For parameter δ=1/3,the truncation error in the temporal direction has 3rd-order accuracy. Finally, numerical examples verify the correctness of the truncation error analysis.
- Burgers equation /
- Hopf-Cole /
- high-order barycentric rational interpolation collocation method /
- generalized trapezoidal formula

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参考文献(33)

Bateman H. Some recent researches on the motion of fluids[J].Monthly Weather Review,1915,43(4): 163-170.

[2]Kreiss G, Kreiss H O. Convergence to steady state of solutions of Burgers’ equation[J].Applied Numerical Mathematics,1986,2(3/4/5): 161-179.

[3]Gurbatov S N, Saichev A I. Probability distribution and spectra of potential hydrodynamic turbulence[J].Radiophysics and Quantum Electronics,1984,27(4): 303-313.

[4]Yepez J. An efficient quantum algorithm for the one-dimensional Burgers equation[PP/OL]. Arxiv (2002-10-11). https://arxiv.org/abs/quant-ph/0210092.

[5]Hammad D A, El-Azab M S. 2N order compact finite difference scheme with collocation method for solving the generalized Burgers’-Huxley and Burgers’-Fisher equations[J].Applied Mathematics and Computation,2015,258: 296-311.

[6]潘悦悦, 杨晓忠. KdV-Burgers方程的一类新本性并行差分格式[J]. 应用数学和力学, 2023,44(5): 583-594. (Pan Yueyue, Yang Xiaozhong. A new class of difference schemes with intrinsic parallelism for the KdV-Burgers equation[J].Applied Mathematics and Mechanics,2023,44(5): 583-594. (in Chinese))

[7]Bressan N, Quarteroni A. An implicit/explicit spectral method for Burgers’ equation[J].Calcolo,1986，〖STHZ〗23(3): 265-284.

[8]王文洽. Burgers方程的一类交替分组方法[J]. 应用数学和力学, 2004,25(2): 213-220.(Wang Wenqia. Class of alternating group method of Burgers’ equation[J].Applied Mathematics and Mechanics,2004,25(2): 213-220. (in Chinese))

[9]Zhu Z M, Chen G, XIE X P. Semi-discrete and fully discrete HDG methods for Burgers’ equation[J].Communications on Pure and Applied Analysis,2023,22(1): 58-81.

[10]郭玲, 王晓丽. Burgers’方程的特征混合有限元数值模拟[J]. 工程数学学报, 2004,21(3): 356-364.(Guo Ling, Wang Xiaoli. A characteristics-mixed finite element method and numerical simulation for Burgers’ equation[J].Chinese Journal of Engineering Mathematics,2004,21(3): 356-364. (in Chinese))

[11]马艳春, 张寅虎, 冯新龙. 二维Burgers方程的RKDG有限元解法[J]. 工程数学学报, 2013,30(3): 442-450.(Ma Yanchun, Zhang Yinhu, Feng Xinlong. The RKDG finite element method for Burgers equation[J].Chinese Journal of Engineering Mathematics,2013,30(3): 442-450. (in Chinese))

[12]Maday Y, Quarteroni A. Legendre and Chebyshev spectral approximations of Burgers’ equation[J].Numerische Mathematik,1981,37(3): 321-332.

[13]安筱, 李珊. 基于广义KdV-Burgers方程的全对角化Chebyshev Dual-Petrov-Galerkin谱方法[J]. 计算机与数字工程, 2025,53(3): 628-631.(An Xiao, Li Shan. Fully diagonalized Chebyshev Dual-Petrov-Galerkin spectral method of generalized KdV-Burgers equation[J].Computer and Digital Engineering,2025,53(3): 628-631. (in Chinese))

[14]董帅, 吴华. 随机系数Burgers方程的广义多项式混沌-谱方法[J]. 应用数学与计算数学学报, 2018,32(3): 675-685.(Dong Shuai, Wu Hua. Generalized polynomial chaos and spectral method for Burgers equation with a random coefficient[J].Communication on Applied Mathematics and Computation,2018,32(3): 675-685. (in Chinese))

[15]Li M, Nikan O, Qiu W, et al. An efficient localized meshless collocation method for the two-dimensional Burgers-type equation arising in fluid turbulent flows[J].Engineering Analysis With Boundary Elements,2022,144: 44-54.

[16]潘伟. 三次MQ径向基函数与Burgers方程求解的研究[D]. 长沙: 中南大学, 2012.(Pan Wei. Researchon the cubic MQ radial basis function and the solution of Burgers equation[D]. Changsha: Central South University, 2012. (in Chinese))

[17]Luo W H, Huang T Z, Gu X M, et al. Barycentric rational collocation methods for a class of nonlinear parabolic partial differential equations[J].Applied Mathematics Letters,2017,68: 13-19.

[18]Wang Y, Chen S, Wu X. A rational spectral collocation method for solving a class of parameterized singular perturbation problems[J].Journal of Computational and Applied Mathematics,2010,233(10): 2652-2660.

[19]王英伟, 陈素琴, 吴雄华. 指数时程差分与有理谱配点法求解奇异摄动Burgers-Huxley问题[J]. 计算数学, 2010,32(2): 171-182.(Wang Yingwei, Chen Suqin, Wu Xionghua. The exponential time differencing and rational spectral collocation method for singularly perturbed Burgers-Huxley problem[J].Mathematica Numerica Sinica,2010,32(2): 171-182. (in Chinese))

[20]Liu L B, Long G, Huang Z, et al. Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer[J].Journal of Computational and Applied Mathematics,2018,335: 312-322.

[21]Zhao Q, Wang R, Wang Z, et al. Barycentric rational collocation method for Burgers’ equation[J].Journal of Mathematics,2022,2022: 2177998.

[22]Liu F, Wang Y, Li S. Barycentric interpolation collocation method for solving the coupled viscous Burgers’ equations[J].International Journal of Computer Mathematics,2018,95(11): 2162-2173.

[23]Hu Y, Peng A, Chen L, et al. Analysis of the barycentric interpolation collocation scheme for the Burgers equation[J].Science Asia,2021,47(6): 758.

[24]Kapoor M, Joshi V. Numerical solution of coupled 1D Burgers’ equation by employing Barycentric Lagrange interpolation basis function based differential quadrature method[J].International Journal for Computational Methods in Engineering Science and Mechanics,2022,23(3): 263-283.

[25]滕宇航, 赖艺颖, 黄浪扬. 应用重心插值的Burgers方程数值解法[J]. 华侨大学学报(自然科学版), 2025,46(1): 104-112.(Teng Yuhang, Lai Yiying, Huang Langyang. Numerical solution method of Burgers equation using barycentric interpolation[J].Journal of Huaqiao University (Natural Science), 2025,46(1): 104-112. (in Chinese))

[26]胡玉蝶, 彭澳, 陈丽权, 等. 有限差分-配点法求解二维Burgers方程[J]. 纯粹数学与应用数学, 2023,39(1): 100-112.(Hu Yudie, Peng Ao, Chen Liquan, et al. A finite difference-collocation method for the Burgers equation[J].Pure and Applied Mathematics,2023,39(1): 100-112. (in Chinese))

[27]Elgindy K T, Dahy S A. High-order numerical solution of viscous Burgers’ equation using a Cole-Hopf barycentric Gegenbauer integral pseudospectral method[J].Mathematical Methods in the Applied Sciences,2018,41(16): 6226-6251.

[28]Shyaman V P, Sreelakshmi A, Awasthi A. Streamlined numerical solutions of Burgers’ equations bridging the tailored finite point method and the Cole Hopf transformation[J].International Journal of Computer Mathematics,2024,101(7): 789-818.

[29]Liao W. A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations[J].International Journal for Numerical Methods in Fluids,2010,64(5): 565-590.

[30]Kannan R, Wang Z J. A high order spectral volume solution to the Burgers’ equation using the Hopf-Cole transformation[J].International Journal for Numerical Methods in Fluids,2012,69(4): 781-801.

[31]Chawla M M, Al-Zanaidi M A, Evans D J. A class of generalized trapezoidal formulas for the numerical integration of[J].International Journal of Computer Mathematics,1996,62(1/2): 131-142.

[32]Chawla M M, Al-Zanaidi M A, Evans D J. Generalized trapezoidal formulas for parabolic equations[J].International Journal of Computer Mathematics,1999,70(3): 429-443.

[33]Li J, Cheng Y. Linear barycentric rational collocation method for solving heat conduction equation[J].Numerical Methods for Partial Differential Equations,2021,37(1): 533-545.

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基于广义梯形公式的二维Burgers方程三阶数值解法

doi: 10.21656/1000-0887.470066

作者简介:
刘陶(1999—)，男,硕士生(E-mail: taoledu@163.com);刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

通讯作者:
刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

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A 3rd-Order Numerical Method for 2D Burgers Equations Based on the Generalized Trapezoid Formula

计量

目录

留言板

基于广义梯形公式的二维Burgers方程三阶数值解法

doi: 10.21656/1000-0887.470066

作者简介: 刘陶(1999—)，男,硕士生(E-mail: taoledu@163.com);刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

通讯作者: 刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

计量

出版历程

A 3rd-Order Numerical Method for 2D Burgers Equations Based on the Generalized Trapezoid Formula

计量

出版历程

目录

作者简介:
刘陶(1999—)，男,硕士生(E-mail: taoledu@163.com);刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).

通讯作者:
刘利斌(1982—),男,教授,博士,硕士生导师(通信作者. E-mail: liulibin969@163.com).