A new high-order compact numerical method for the two-dimensional nonlinear Burgers’ equation

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

The Burgers’ equation is a key model equation in fluid mechanics and nonlinear physics, widely used to describe the formation and propagation of shock waves. This paper proposes a highly accurate numerical algorithm to solve the two-dimensional Burgers’ equation. In the algorithmic framework, the time derivative is discretized using a fourth-order backward finite difference approach. In contrast, the spatial derivatives are processed by a fourth-order Padé scheme and a fourth-order compact finite difference scheme, respectively. To linearize the nonlinear term in the equation, an extrapolation-based linearization technique is introduced, which effectively reduces the computational complexity while maintaining the numerical accuracy. Theoretical analyses of this algorithm demonstrated its unique solvability and stability. Numerous numerical experiments verify the scheme’s stability, computational efficiency, and convergence. This approach achieves a fourth-order accuracy in both the temporal and spatial dimensions, thus offering the benefit of maintaining high agreement between the numerical and exact solutions, even when using relatively large time steps, thereby significantly improving the computational efficiency. In addition, the algorithm employs linearization to ease the computational demands, thereby providing a viable and efficient scheme to address high-dimensional nonlinear Burgers’ equations.

Keywords

two-dimensional Burgers’ equation / fourth-order compact finite difference scheme / fourth-order backward finite difference formula / fourth-order Padé scheme

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] K. T. Elgindy, High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures, Appl. Numer. Math., 113 (2017), 1-25. https://doi.org/10.1016/j.apnum.2016.10.014

[2] K. Fujisawa, A. Asada, Nonlinear parametric sound enhancement through different fluid layer and its application to noninvasive measurement, Measurement, 94 (2016), 726-733. https://doi.org/10.1016/j.measurement.2016.09.004

[3] Y. D. Jin, J. Zhou, Z. K. Shi, Lattice hydrodynamic model for traffic flow on curved road with passing, Nonlinear Dyn., 89 (2017), 107-124. https://doi.org/10.1007/s11071-017-3439-8

[4] K. Shah, Solution of Burgers’ equation in a one-dimensional groundwater recharge by spreading using q-Homotopy analysis method, Eur. J. Pure Appl. Math., 9 (2016), 114-124. https://www.ejpam.com/index.php/ejpam

[5] L. Yang, X. Pu, Derivation of the Burgers’ equation from the gas dynamics, Commun. Math. Sci., 14 (2016), 671-682. https://doi.org/10.4310/cms.2016.v14.n3.a4

[6] A. Khoshfetrat, M. Abedini, A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burger’s equations in irregular domain, Appl. Math. Modell., 36 (2012), 1885-1901. https://doi.org/10.1016/j.apm.2011.07.079

[7] R. Mohanty, N. Setia, A new high accuracy two-level implicit off-step discretization for the system of two space dimensional quasi-linear parabolic partial differential equations, Appl. Math. Comput., 219 (2012), 2680-2697. https://doi.org/10.1016/j.amc.2012.08.100

[8] M. Seydaoğlu, U. Erdoğan, T. Özış, Numerical solution of Burgers’ equation with high order splitting methods, J. Comput. Appl. Math., 291 (2016), 410-421. https://doi.org/10.1016/j.cam.2015.04.021

[9] R. C. Mittal, R. Rohila, A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method, J. Math. Chem., 55 (2017), 673-695. https://doi.org/10.1007/s10910-016-0703-y

[10] I. A. Ganaie, V. K. Kukreja, Numerical solution of Burgers’ equation by cubic Hermite collocation method, Appl. Math. Comput., 237 (2014), 571-581. https://doi.org/10.1016/j.amc.2014.03.102

[11] X. J. Yang, Y. B. Ge, L. Zhang, A class of high-order compact difference schemes for solving the Burgers’ equations, Appl. Math. Comput., 358 (2019), 394-417. https://doi.org/10.1016/j.amc.2019.04.023

[12] V. Mukundan, A. Awasthi, Linearized implicit numerical method for Burgers’ equation, Nonlinear Eng., 5 (2016), 219-234. https://doi.org/10.1515/nleng-2016-0031

[13] T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, Higher-order accurate numerical solution of unsteady Burgers’ equation, Appl. Math. Comput., 250 (2015), 701-707. https://doi.org/10.1016/j.amc.2014.11.013

[14] X. Miao, J. Gan, T. Ma, L. Wu, A linear time-space high-order compact finite difference scheme for solving the one-dimensional nonlinear Burgers’ equation, J. Appl. Anal. Comput., 15 (2025), 3226-3245. https://doi.org/10.11948/20240527

[15] A. Korkmaz, I. Dag, Cubic B-spline differential quadrature methods and stability for Burgers’ equation, Eng. Comput., 30 (2013), 320-344. https://doi.org/10.1108/02644401311314312

[16] G. Arora, B. Singh, Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method, Appl. Math. Comput., 224 (2013), 166-177. https://doi.org/10.1016/j.amc.2013.08.071

[17] H. Shukla, M. Tamsir, V. Srivastava, J. Kumar, Numerical solution of two dimensional coupled viscous Burgers’ equation using modified cubic B-spline differential quadrature method, AIP Adv., 4 (2014), 117134. https://doi.org/10.1063/1.4902507

[18] R. Mittal, A. Tripathi, A. Ali, Numerical solutions of two-dimensional Burgers’ equations using modified B-cubic B-spline finite elements, Eng. Comput., 32 (2015), 1275-1306. https://doi.org/10.1108/ec-04-2014-0067

[19] R. C. Mittal, R. K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic b-splines collocation method, Appl. Math. Comput., 218 (2012), 7839-7855. https://doi.org/10.1016/j.amc.2012.01.059

[20] C. Fan, P. Li, Generalized finite difference method for solving two-dimensional Burgers’ equations, Procedia Eng., 79 (2014), 55-60. https://doi.org/10.1016/j.proeng.2014.06.310

[21] R. C. Mittal, R. Jiwari, Differential quadrature method for two-dimensional Burgers’ equations, Int. J. Comput. Methods Eng. Sci. Mech., 10 (2009), 450-459. https://doi.org/10.1080/15502280903111424

[22] A. A. Soliman, On the solution of two-dimensional coupled Burgers’ equations by variational iteration method, Chaos, Solitons Fractals, 40 (2009), 1146-1155. https://doi.org/10.1016/j.chaos.2007.08.069

[23] H. Zhu, H. Shu, M. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete adomian decomposition method, Comput. Math. Appl., 60 (2010), 840-848. https://doi.org/10.1016/j.camwa.2010.05.031

[24] G. Zhao, X. Yu, R. Zhang, The new numerical method for solving the system of two-dimensional Burgers’ equations, Comput. Math. Appl., 62 (2011), 3279-3291. https://doi.org/10.1016/j.camwa.2011.08.044

[25] S. Baharlouei, R. Mokhtari, N. Chegini, Solving two-dimensional coupled Burgers’ equations via a stable hybridized discontinuous Galerkin method, Iran. J. Numer. Anal. Optim., 13 (2023), 397-425. https://doi.org/10.22067/ijnao.2023.80916.1215

[26] A. R. Bahadır, A fully implicit finite-difference scheme for two-dimensional Burgers equations, Appl. Math. Comput., 137 (2003), 131-137. https://doi.org/10.1016/s0096-3003(02)00091-7

[27] X. Chen, Y. Chen, A weak Galerkin finite element method for the two-dimensional Burgers’ equation, Comput. Math. Appl., 196 (2025), 84-103. https://doi.org/10.1016/j.camwa.2025.07.010

[28] S. Gottlieb, C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers’ equation, J. Sci. Comput., 53 (2012), 102-128. https://doi.org/10.1007/s10915-012-9621-8

[29] X. J. Yang, Y. B. Ge, B. Lan, A class of compact finite difference schemes for solving the 2D and 3D Burgers’ equations, Math. Comput. Simul., 185 (2021), 510-524. https://doi.org/10.1016/j.matcom.2021.01.009

[30] X. Peng, D. Xu, W. Qiu, Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation, Math. Comput. Simul., 208 (2023), 702-726. https://doi.org/10.1016/j.matcom.2023.02.004

[31] A. K. Mittal, A space-time pseudospectral method for solving multi-dimensional quasi-linear parabolic partial differential (Burgers’) equations, Appl. Numer. Math., 195 (2024), 39-53. https://doi.org/10.1016/j.apnum.2023.09.005

[32] Y. Sun, Q. Chen, T. Chen, L. Yong, Explicit numerical manifold characteristic Galerkin method for solving Burgers’ equation, Axioms, 13 (2024), 343. https://doi.org/10.3390/axioms13060343

[33] X. Zhang, Y. Zhang, Q. Jin, Y. Jin, Two sixth-order, L^∞ convergent, and stable compact difference schemes for the generalized Rosenau-KdV-RLW equation , J. Comput. Appl. Math., 459 (2025), 116382. https://doi.org/10.1016/j.cam.2024.116382

[34] A. Polwang, K. Poochinapan, B. Wongsaijai, Numerical simulation of wave flow : Integrating the BBM-KdV equation using compact difference schemes, Math. Comput. Simul., 236 (2025), 70-89. https://doi.org/10.1016/j.matcom.2025.03.012

[35] S. D. Wang, T. F. Ma, L. L. Wu, X. J. Yang, Two high-order compact finite difference schemes for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation, Appl. Math. Comput., 496 (2025), 129360. https://doi.org/10.1016/j.amc.2025.129360

[36] S. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42. https://doi.org/10.1016/0021-9991(92)90324-r

[37] Y. Kang, H. Liao, Energy stability of BDF methods up to fifth-order for the molecular beam epitaxial model without slope selection, J. Sci. Comput., 91 (2022), 47. https://doi.org/10.1007/s10915-022-01830-x