Numerical solution of a coupled Burgers' equation via barycentric interpolation collocation method

Xiumin Lyu; Jin Li; Wanjun Song; Xiumin Lyu; Jin Li; Wanjun Song

doi:10.3934/era.2025070

Electronic Research Archive

2025, Volume 33, Issue 3: 1490-1509. doi: 10.3934/era.2025070

Previous Article Next Article

Research article Special Issues

Numerical solution of a coupled Burgers' equation via barycentric interpolation collocation method

1.
School of Science, Shandong Jiaotong University, Jinan 250357, China
2.
School of Science, Shandong Jianzhu University, Jinan 250101, China

Received: 19 January 2025 Revised: 02 March 2025 Accepted: 06 March 2025 Published: 17 March 2025

In this study, the numerical solution of a two-dimensional coupled Burgers' equation is investigated by the barycentric interpolation collocation method. The spatial-temporal domain of the equation is made discrete by employing barycentric interpolation, and a corresponding differentiation matrix based on this interpolation technique is constructed. The effectiveness and high accuracy of the proposed method are demonstrated for solving the two-dimensional coupled Burgers' equation.
- numerical calculation,
- barycentric collocation method,
- Burgers' equation,
- error distribution
Citation: Xiumin Lyu, Jin Li, Wanjun Song. Numerical solution of a coupled Burgers' equation via barycentric interpolation collocation method[J]. Electronic Research Archive, 2025, 33(3): 1490-1509. doi: 10.3934/era.2025070

Related Papers:

Abstract

In this study, the numerical solution of a two-dimensional coupled Burgers' equation is investigated by the barycentric interpolation collocation method. The spatial-temporal domain of the equation is made discrete by employing barycentric interpolation, and a corresponding differentiation matrix based on this interpolation technique is constructed. The effectiveness and high accuracy of the proposed method are demonstrated for solving the two-dimensional coupled Burgers' equation.

References

[1]	M. P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan, V. S. Aswin, A systematic literature review of Burgers' equation with recent advances, Pramana, 90 (2018), 69. https://doi.org/10.1007/s12043-018-1559-4 doi: 10.1007/s12043-018-1559-4
[2]	M. Khan, A novel solution technique for two dimensional Burger's equation, Alexandria Eng. J., 53 (2014), 485–490. https://doi.org/10.1016/j.aej.2014.01.004 doi: 10.1016/j.aej.2014.01.004
[3]	R. Jiwari, R. C. Mittal, K. K. Sharma, A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers' equation, Appl. Math. Comput., 219 (2013), 6680–6691. https://doi.org/10.1016/j.amc.2012.12.035 doi: 10.1016/j.amc.2012.12.035
[4]	M. Kumar, S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers' equation, Comput. Phys. Commun., 185 (2014), 809–817. https://doi.org/10.1016/j.cpc.2013.11.012 doi: 10.1016/j.cpc.2013.11.012
[5]	M. Tamsir, V. K. Srivastava, R. Jiwari, An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers' equation, Appl. Math. Comput., 290 (2016), 111–124. https://doi.org/10.1016/j.amc.2016.05.048 doi: 10.1016/j.amc.2016.05.048
[6]	S. Bak, P. Kim, D. Kim, A semi-Lagrangian approach for numerical simulation of coupled Burgers' equations, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 31–44. https://doi.org/10.1016/j.cnsns.2018.09.007 doi: 10.1016/j.cnsns.2018.09.007
[7]	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 doi: 10.1016/j.chaos.2007.08.069
[8]	H. Aminikhah, An analytical approximation for coupled viscous Burgers' equation, Appl. Math. Modell., 37 (2013), 5979–5983. https://doi.org/10.1016/j.apm.2012.12.013 doi: 10.1016/j.apm.2012.12.013
[9]	H. Eltayeb, I. Bachar, A note on singular two-dimensional fractional coupled Burgers' equation and triple Laplace Adomian decomposition method, Bound. Value Probl., 2020 (2020), 129. https://doi.org/10.1186/s13661-020-01426-0 doi: 10.1186/s13661-020-01426-0
[10]	R. K. Alhefthi, H. Eltayeb, The Solution of two-dimensional coupled Burgers' equation by G-double laplace transform, J. Funct. Spaces, 2023 (2023), 4320612. https://doi.org/10.1155/2023/4320612 doi: 10.1155/2023/4320612
[11]	D. Kaya, An explicit solution of coupled viscous Burgers' equation by the decomposition method, Int. J. Math. Math. Sci., 27 (2001), 675–680. https://doi.org/10.1155/S0161171201010249 doi: 10.1155/S0161171201010249
[12]	J. Biazar, H. Ghazvini, Exact solutions for nonlinear Burgers' equation by homotopy perturbation method, Numer. Methods Partial Differ. Equations, 25 (2009), 833–842. https://doi.org/10.1002/num.20376 doi: 10.1002/num.20376
[13]	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 doi: 10.1016/S0096-3003(02)00091-7
[14]	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 doi: 10.1016/j.camwa.2011.08.044
[15]	E. Hopf, The partial differential equation $u_{t}+uu_{x} = u_xx$, Commun. Pure Appl. Math., 3 (1950), 201–230. https://doi.org/10.1002/cpa.3160030302 doi: 10.1002/cpa.3160030302
[16]	J. D. Cole, On a quasilinear parabolic equations occurring in aerodynamics, Q. Appl. Math., 9 (1951), 225–236. https://doi.org/10.1090/qam/42889 doi: 10.1090/qam/42889
[17]	C. A. J. Fletcher, Generating exact solutions of the two-dimensional Burgers' equations, Int. J. Numer. Methods Fluids, 3 (1983), 213–216. https://doi.org/10.1002/fld.1650030302 doi: 10.1002/fld.1650030302
[18]	Y. Chai, J. Ouyang, Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers' equations at high Reynolds numbers, Comput. Math. Appl., 79 (2020), 1287–1301. https://doi.org/10.1016/j.camwa.2019.08.036 doi: 10.1016/j.camwa.2019.08.036
[19]	A. J. Hussein, H. A. Kashkool, A weak Galerkin finite element method for two-dimensional coupled Burgers' equation by using polynomials of order ($k, k-1, k-1$), J. Interdiscip. Math., 23 (2020), 777–790. https://doi.org/10.1080/09720502.2019.1706844 doi: 10.1080/09720502.2019.1706844
[20]	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 doi: 10.1016/j.matcom.2023.02.004
[21]	Y. Uçar, N. M. Yağmurlu, M. K. Yiğit, Numerical Solution of the coupled Burgers equation by trigonometric B-spline collocation method, Math. Methods Appl. Sci., 46 (2023), 6025–6041. https://doi.org/10.1002/mma.8887 doi: 10.1002/mma.8887
[22]	Z. Dehghan, J. Rashidinia, Numerical solution of coupled viscous Burgers' equations using RBF-QR method, Math. Sci., 17 (2023), 317–324. https://doi.org/10.1007/s40096-022-00472-2 doi: 10.1007/s40096-022-00472-2
[23]	S. Park, Y. Jeon, P. Kim, S. Bak, An error predict-correction formula of the load vector in the BSLM for solving three-dimensional Burgers' equations, Math. Comput. Simul., 221 (2024), 222–243. https://doi.org/10.1016/j.matcom.2024.03.001 doi: 10.1016/j.matcom.2024.03.001
[24]	S. Torkaman, M. Heydari, G. B. Loghmani, Piecewise barycentric interpolating functions for the numerical solution of Volterra integro-differential equations, Math. Methods Appl. Sci., 45 (2022), 6030–6061. https://doi.org/10.1002/mma.8154 doi: 10.1002/mma.8154
[25]	H. Liu, J. Huang, Y. Pan, J. Zhang, Barycentric interpolation collocation method for solving linear and nonlinear high-dimensional Fredholm integral equations, J. Comput. Appl. Math., 327 (2018), 141–154. https://doi.org/10.1016/j.cam.2017.06.004 doi: 10.1016/j.cam.2017.06.004
[26]	J. Li, Y. Cheng, Linear barycentric rational collocation method for solving heat conduction equation, Numer. Methods Partial Differ. Equations, 37 (2021), 533–545. https://doi.org/10.1002/num.22539 doi: 10.1002/num.22539
[27]	J. Li, Linear barycentric rational collocation method to solve plane elasticity problems, Math. Biosci. Eng., 20 (2023), 8337–8357. https://doi.org/10.3934/mbe.2023365 doi: 10.3934/mbe.2023365
[28]	J. Li, Barycentric rational collocation method for semi-infinite domain problems, AIMS Math., 8 (2023), 8756–8771. https://doi.org/10.3934/math.2023439 doi: 10.3934/math.2023439
[29]	S. Li, Z. Wang, Meshless Barycentric Interpolation Collocation Method-Algorithmics, Programs & Applications in Engineering, Science Publishing, Beijing, 2012.
[30]	Z. Wang, S. Li, Barycentric Interpolation Collocation Method for Nonlinear Problems, National Defense Industry Press, Beijing, 2015.
[31]	Ö. Oruç, Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation, Int. J. Comput. Math., 98 (2021), 461–479. https://doi.org/10.1080/00207160.2020.1755432 doi: 10.1080/00207160.2020.1755432
[32]	Y. Hu, A. Peng, L. Chen, Y. Tong, Z. Weng, A finite difference-collocation method for the Burgers equation, Pure Appl. Math., 39 (2023), 100–112.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)