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

Liu Tao; Liu Libin

doi:10.21656/1000-0887.470066

Volume 47 Issue 9

Sep. 2026

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Liu Tao, Liu Libin. A 3rd-Order Numerical Method for 2D Burgers Equations Based on the Generalized Trapezoid Formula[J]. Applied Mathematics and Mechanics, 2026, 47(9): 1197-1210. doi: 10.21656/1000-0887.470066

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

doi: 10.21656/1000-0887.470066

Liu Tao,
Liu Libin^,

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

Received Date: 2026-03-17
Publish Date: 2026-09-01

Abstract

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

(Recommended by Liu Huanwen, Member of the Editorial Board of AMM)

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References(33)

References

[1]	Bateman H. Some recent researches on the motion of fluids[J]. Monthly Weather Review, 1915, 43(4): 163-170. doi: 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2
[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. doi: 10.1007/BF01036611
[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. doi: 10.1016/j.amc.2015.02.009
[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. doi: 10.21656/1000-0887.430128 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) doi: 10.21656/1000-0887.430128
[7]	Bressan N, Quarteroni A. An implicit/explicit spectral method for Burgers' equation[J]. Calcolo, 1986, 23(3): 265-284. doi: 10.1007/BF02576532
[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. doi: 10.3934/cpaa.2021132
[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. doi: 10.1007/BF01400311
[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. doi: 10.1016/j.enganabound.2022.08.007
[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. doi: 10.1016/j.aml.2016.12.011
[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. doi: 10.1016/j.cam.2009.11.011
[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. doi: 10.1016/j.cam.2017.12.012
[21]	Zhao Q, Wang R, Wang Z, et al. Barycentric rational collocation method for Burgers' equation[J]. Journal of Mathematics, 2022, 2022: 2177998. doi: 10.1155/2022/2177998
[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. doi: 10.1080/00207160.2017.1384546
[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. doi: 10.2306/scienceasia1513-1874.2021.081
[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. doi: 10.1080/15502287.2021.1954726
[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. doi: 10.1002/mma.5135
[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. doi: 10.1080/00207160.2024.2384598
[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. doi: 10.1002/fld.2163
[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. doi: 10.1002/fld.2612
[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. doi: 10.1080/00207169908804765
[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. doi: 10.1002/num.22539