The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.
| Published in | Pure and Applied Mathematics Journal (Volume 13, Issue 2) |
| DOI | 10.11648/j.pamj.20241302.11 |
| Page(s) | 17-28 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Burgers’ Equation, Hopf-Cole Transformation, Explicit Scheme, Parallel Scalability
| [1] | M. P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan, and V. S. Aswin, “A systematic literature review of burgers equation with recent advances, Pramana, vol. 90, pp. 1-21, 2018. https://doi.org/10.1007/s12043-018-1559-4 |
| [2] | M. Abdullah, M. Yaseen, and M. De la Sen, “An efficient collocation method based on Hermite formula and cubic B-splines for numerical solution of the Burgers equation, Math. Comput. Simulation, vol. 197, no. C, pp. 166-184, 2022. https://doi.org/10.1016/j.matcom.2022.02 |
| [3] | A. Dogan, “A galerkin finite element approach to burgers equation, Appl. Math. Comput., vol. 157, no. 2, pp. 331- 346, 2004. https://doi.org/10.1016/j.amc.2003.08.037 |
| [4] | I. Ganaie and V. Kukreja, “Numerical solution of burgers equation by cubic hermite collocation method, Appl. Math. Comput., vol. 237, pp. 571-581, 2014. https://doi.org/10.1016/j.amc.2014.03.102 |
| [5] | Y. Hon and X. Mao, “An efficient numerical scheme for burgers equation, Appl. Math. Comput., vol. 95, no. 1, pp. 37-50, 1998. https://doi.org/10.1016/S0096-3003(97)10060-1 |
| [6] | P. Huang and A. Abduwali, “The modified local cranknicolson method for one- and two- dimensional burgers equations, Comput. Math. Appl., vol. 59, no. 8, pp. 2452-2463, 2010. https://doi.org/10.1016/j.camwa.2009.08.069 |
| [7] | A. Vs and A. Awasthi, “A differential quadrature based numerical method for highly accurate solutions of burgers equation: Dqm based numerical method for burgers equation, Numer. Meth. Part. D. E., vol. 33, 07 2017. https://doi.org/10.1002/num.22178 |
| [8] | X. Yang, Y. Ge, and B. Lan, “A class of compact finite difference schemes for solving the 2d and 3d burgers equations, Math. Comput. Simulation, vol. 185, pp. 510-534, 2021. https://doi.org/10.1016/j.matcom.2021.01.009 |
| [9] | Y. Guo, Y. feng Shi, and Y. min Li, “A fifth- order finite volume weighted compact scheme for solving one-dimensional burgers equation, Appl. Math. Comput., vol. 281, pp. 172-185, 2016. https://doi.org/10.1016/j.amc.2016.01.061 |
| [10] | S. Gupta and V. K. Kukreja, “An improvised collocation algorithm with specific end conditions for solving modified burgers equation, Numer. Meth. Part. D. E., vol. 37, no. 1, pp. 874-896, 2021. https://doi.org/10.1002/num.22557 |
| [11] | S. R. Jena and G. S. Gebremedhin, “Decatic b-spline collocation scheme for approximate solution of burgers equation, Numer. Meth. Part. D. E., vol. 39, no. 3, pp. 1851-1869, 2023. https://doi.org/10.1002/num.22747 |
| [12] | Y. Jiang, X. Chen, R. Fan, and X. Zhang, “High order semi-implicit weighted compact nonlinear scheme for viscous burgers equations, Math. Comput. Simulation, vol. 190, pp. 607-621, 2021. https://doi.org/10.1016/j.matcom.2021.06.006 |
| [13] | R. K. Mohanty and J. Talwar, “Anewcompactalternating group explicit iteration method for the solution of nonlinear time-dependent viscous burgers equation, Numer. Anal. Appl., vol. 8, pp. 314-328, 2015. https://doi.org/10.1134/S1995423915040059 |
| [14] | R. Zhang, Y. Xi-Jun, and Z. Guo-Zhong, “Local discontinuous galerkin method for solving burgers and coupled burgers equations, Chin. Phys. B, vol. 20, no. 11, p. 110205, 11 2011. https://doi.org/10.1088/1674-1056/20/11/110205 |
| [15] | M. K. Kadalbajoo and A. Awasthi, “A numerical method based on crank-nicolson scheme for burgers equation, Appl. Math. Comput., vol. 182, no. 2, pp. 1430-1442, 2006. https://doi.org/10.1016/j.amc.2006.05.030 |
| [16] | R. Kannan and Z. Wang, “A high order spectral volume solution to the burgers equation using the hopfcole transformation, Internat. J. Numer. Methods Fluids, vol. 69, no. 4, pp. 781-801, 2012. https://doi.org/10.1002/fld.2612 |
| [17] | S. S. Kumbhar and S. Thakar, “Galerkin finite element method for forced burgers equation, J. Math. Model., vol. 7, no. 4, pp. 445-467, 2019. https://doi.org/10.22124/jmm.2019.13259.1265 |
| [18] | S. Kutluay, A. Bahadir, and A. zde, “Numerical solution of one-dimensional burgers equation: explicit and exact- explicit finite difference methods, J. Comput. Appl. Math., vol. 103, no. 2, pp. 251-261, 1999. https://doi.org/10.1016/S0377-0427(98)00261-1 |
| [19] | W. Liao, “An implicit fourth-order compact finite difference scheme for one-dimensional burgers equation, Appl. Math. Comput., vol. 206, no. 2, pp. 755-764, 2008. https://doi.org/10.1016/j.amc.2008.09.037 |
| [20] | V. Mukundan and A. Awasthi, “Efficient numerical techniques for burgers equation, Appl. Math. Comput., vol. 262, pp. 282-297, 2015. https://doi.org/10.1016/j.amc.2015.03.122 |
| [21] | K. Pandey, L. Verma, and A. K. Verma, “On a finite difference scheme for burgers equation, Appl. Math. Comput., vol. 215, no. 6, pp. 2206-2214, 2009. https://doi.org/10.1016/j.amc.2009.08.018 |
| [22] | K. Sakai and I. Kimura, “A numerical scheme based on a solution of nonlinear advectiondiffusion equations, J. Comput. Appl. Math., vol. 173, no. 1, pp. 39-55, 2005. https://doi.org/10.1016/j.cam.2004.02.019 |
| [23] | S.-S. Xie, S. Heo, S. Kim, G. Woo, and S. Yi, “Numerical solution of one-dimensional burgers equation using reproducing kernel function, J. Comput. Appl. Math., vol. 214, no. 2, pp. 417-434, 2008. https://doi.org/10.1016/j.cam.2007.03.010 |
| [24] | G. Zhao, X. Yu, and R. Zhang, “The new numerical method for solving the system of two- dimensional burgers equations, Comput. Math. Appl., vol. 62, no. 8, pp. 3279-3291, 2011. https://doi.org/10.1016/j.camwa.2011.08.044 |
| [25] | E. Aksan, “A numerical solution of burgers equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., vol. 170, no. 2, pp. 895-904, 2005. https://doi.org/10.1016/j.amc.2004.12.027 |
| [26] | J. Caldwell and P. Smith, “Solution of burgers equation with a large reynolds number, Appl. Math. Model., vol. 6, no. 5, pp. 381-385, 1982. https://doi.org/10.1016/S0307- 904X(82)80102-9 |
| [27] | J. Caldwell, P. Wanless, and A. Cook, “A finite element approach to burgers equation, Appl. Math. Model., vol. 5, no. 3, pp. 189-193, 1981. https://doi.org/10.1016/0307- 904X(81)90043-3 |
| [28] | Y. Chai and J. Ouyang, “Appropriate stabilized galerkin approaches for solving two-dimensional coupled burgers equations at high reynolds numbers, Comput. Math. Appl., vol. 79, no. 5, pp. 1287-1301, 2020. https://doi.org/10.1016/j.camwa.2019.08.036 |
| [29] | G. Arora and B. K. Singh, “Numerical solution of burgersequationwithmodifiedcubicb- splinedifferential quadrature method, Appl. Math. Comput., vol. 224, pp. 166-177, 2013. https://doi.org/10.1016/j.amc.2013.08.071 |
| [30] | M. Ghasemi, “An efficient algorithm based on extrapolation for the solution of nonlinear parabolic equations, Int. J. Nonlinear Sci. Numer. Simul., vol. 19, no. 1, pp. 37-51, 2018. https://doi.org/10.1515/ijnsns-2017-0060 |
| [31] | B. K. Singh and M. Gupta, “A new efficient fourth order collocation scheme for solving burgers equation, Appl. Math. Comput., vol. 399, p. 126011, 2021. https://doi.org/10.1016/j.amc.2021.126011 |
| [32] | M. Tamsir, N. Dhiman, and V. K. Srivastava, “Extended modified cubic b-spline algorithm for nonlinear burgers equation, Beni-Suef Univ. J. Basic Appl. Sci., vol. 5, no. 3, pp. 244-254, 2016. https://doi.org/10.1016/j.bjbas.2016.09.001 |
| [33] | Y. Gao, L.-H. Le, and B.-C. Shi, “Numerical solution of burgers equation by lattice boltzmann method, Appl. Math. Comput., vol. 219, no. 14, pp. 7685-7692, 2013. https://doi.org/10.1016/j.amc.2013.01.056 |
| [34] | N. Kumar, R. Majumdar, and S. Singh, “Predictorcorrector nodal integral method for simulation of high reynolds number fluid flow using larger time steps in burgers equation, Comput. Math. Appl., vol. 79, no. 5, pp. 1362-1381, 2020. https://doi.org/10.1016/j.camwa.2019.09.001 |
| [35] | F. M. de Lara and E. Ferrer, “Accelerating high order discontinuous galerkin solvers using neural networks: 1d burgers equation, Comput. & Fluids, vol. 235, p. 105274, 2022. https://doi.org/10.1016/j.compfluid.2021.105274 |
| [36] | R. Bridson, Fluid Simulation for Computer Graphics, Second Edition. Taylor & Francis, 2015. ISBN 9781482232837. |
| [37] | P. Olver, Introduction to Partial Differential Equations, ser. Undergraduate Texts in Mathematics. Springer International Publishing, 2013. ISBN 9783319020990. |
| [38] | M. Stone and P. Goldbart, Mathematics for Physics: A Guided Tour for Graduate Students. Cambridge University Press, 2009. ISBN 9780521854030. |
| [39] | M. Sarboland and A. Aminataei, “On the numerical solution of one-dimensional nonlinear nonhomogeneous burgers equation, J. Appl. Math., vol. 2014, pp. 598432: 1-598432: 15, 2014. https://doi.org/10.1155/2014/598432 |
| [40] | E. R. Bentom and G. W. Platzman, “A table of solutions of the one-dimensional burgers equation, Quart. Appl. Math., vol. 30, no. 2, pp. 195-212, 1972. https://doi.org/10.1090/qam/306736 |
| [41] | J. Ramos, “Picards iterative method for nonlinear advectionreactiondiffusion equations, Appl. Math. Comput., vol. 215, no. 4, pp. 1526-1536, 2009. https://doi.org/10.1016/j.amc.2009.07.004 |
APA Style
Kanoksirirath, S. (2024). An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure and Applied Mathematics Journal, 13(2), 17-28. https://doi.org/10.11648/j.pamj.20241302.11
ACS Style
Kanoksirirath, S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl. Math. J. 2024, 13(2), 17-28. doi: 10.11648/j.pamj.20241302.11
AMA Style
Kanoksirirath S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl Math J. 2024;13(2):17-28. doi: 10.11648/j.pamj.20241302.11
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Download@article{10.11648/j.pamj.20241302.11,
author = {Somrath Kanoksirirath},
title = {An Integral-like Numerical Approach for Solving Burgers’ Equation},
journal = {Pure and Applied Mathematics Journal},
volume = {13},
number = {2},
pages = {17-28},
doi = {10.11648/j.pamj.20241302.11},
url = {https://doi.org/10.11648/j.pamj.20241302.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241302.11},
abstract = {The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.},
year = {2024}
}
TY - JOUR T1 - An Integral-like Numerical Approach for Solving Burgers’ Equation AU - Somrath Kanoksirirath Y1 - 2024/06/12 PY - 2024 N1 - https://doi.org/10.11648/j.pamj.20241302.11 DO - 10.11648/j.pamj.20241302.11 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 17 EP - 28 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20241302.11 AB - The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction. VL - 13 IS - 2 ER -
National Electronics and Computer Technology Center (NECTEC), National Science and Technology Development Agency (NSTDA), Pathum Thani, Thailand
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