Abstract
In this article, we propose semi-approximate approach in finding a solution of Burgers’ equation which is one of the partial differential equations (PDEs). Without using the Newton method for linearization, we derive the approximation equation of the proposed problem by using second-order implicit scheme together with the semi-approximate approach. Then this approximation equation leads a huge scale and sparse linear system. Having this linear system, the Successive Overrelaxation (SOR) iteration will be performed as a linear solver. The formulation and execution of SOR iteration are included in this paper. This paper proposed four examples of Burgers’ equations to determine the performance of the suggested method. The test results discovered that the SOR iteration is more effective than GS iteration with less time of execution and minimum iteration numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kutluay S, Esen A, Dag I (2004) Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J Comput Appl Math 167:21–33
Mohamed NA (2019) Solving one-and two-dimensional unsteady Burgers’ equation using fully implicit finite difference schemes. Arab J Basic Appl Sci 26(1):254–268
Guo Y, Shi Y, Li Y (2016) A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation. Appl Math Comput 281:172–185
Mohamed NA (2018) Fully implicit scheme for solving Burgers’ equation based on finite difference method. Egypt Int J Eng Sci Tech 26:38–44
Chen Y, Zhang T (2019) A weak galerkin finite element method for Burgers’ equation. J Comput Appl Math 348:103–119
Zhang Y, Lin J, Reutskiy S, Sun H, Feng W (2020) The improved backward substitution method for the simulation of time-dependent nonlinear coupled Burgers’ equations. Results Phys 18
Rahman K, Helil N, Yimin R (2010) Some new semi-implicit finite difference schemes for numerical solution of Burgers equation. In: 2010 International conference on computer application and system modelling, vol 14, pp 451–455
Ozis T, Aslan Y (2005) The semi-approximate approach for solving Burgers’ equation with high reynolds number. Appl Maths Comp 163:131–145
Liao W (2008) An implicit fourth-order compact finite-difference scheme for one-dimensional Burgers’ equation. Appl Math Comput 206:755–764
Young DM (1954) Iterative methods for solving partial difference equations of elliptic type. Trans Am Math Soc 76(1):92–111
Young DM (1970) Convergence properties of the symmetric and unsymmetric successive overrelaxation methods and related methods. Math Comput 24(112):798–807
Sunarto A, Sulaiman J, Saudi A (2014) Full-sweep SOR iterative method to solve space-fractional diffusion equations. Aust J Basic Appl Sci 8(24):153–158
Zainal NFA, Sulaiman J, Alibubin MU (2018) Application of SOR iteration with nonlocal arithmetic discretization scheme for solving burger’s equation. In: AIP Conference Proceedings, vol 2013, no 1, p 020035
Sari M, Tunc H, Seydaoglu M (2019) Higher order splitting approaches in analysis of the Burgers equation. Kuwait J Sci 46(1):1–14
Zainal NFA, Sulaiman J, Alibubin MU (2019) Application of half-sweep SOR iteration with nonlocal arithmetic discretization scheme for solving Burgers’ equation. ARPN J Eng Appl Sci 14(3):616–621
Muhiddin FA, Sulaiman J, Sunarto A (2019) Grunwald implicit solution for solving one-dimensional time-fractional parabolic equations using SOR iteration. J Phys Conf Ser 1358
Ali NAM, Rahman R, Sulaiman J, Ghazali K (2019) Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration. Indonesian J Electr Eng Comput Sci 16:1430–1438
Mohamad NS, Sulaiman J (2019) The piecewise collocation solution of second kind Fredholm by using quarter-sweep iteration. J Phys Conf Ser 1358
Biazar J, Aminikhah H (2009) Exact and numerical solutions for non-linear Burger’s equation by VIM. Math Comput Model 49:1394–1400
Tamsir M, Dhiman N, Srivastava VK (2016) Extended modified cubic B-spline algorithm for nonlinear Burger’s equation. Beni-Suef Univ J Basic Appl Sci 5:244–254
Arora G, Joshi V (2017) A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers equation in one and two dimensions. Alex Eng J
Raslan KR (2003) A collocation solution for Burgers equation using quadratic B-spline finite elements. Int J Comput Math 80:931–938
Evans DJ (1985) Group explicit iterative methods for solving large linear systems. Int J Comput Math 17(1):81–108
Zainal NFA, Sulaiman J, Alibubin MU (2019b) Application of four-point EGSOR iteration with nonlocal arithmetic mean discretization scheme for solving Burger’s equation. J Phys Conf Ser 1358:012051
Acknowledgements
The authors thankfully recognized that this paper was mainly financed by Postgraduate Centre from Universiti Malaysia Sabah, Kota Kinabalu, Malaysia.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Zainal, N.F.A., Sulaiman, J., Saudi, A., Ali, N.A.M. (2021). Semi-approximate Solution for Burgers’ Equation Using SOR Iteration. In: Alfred, R., Iida, H., Haviluddin, H., Anthony, P. (eds) Computational Science and Technology. Lecture Notes in Electrical Engineering, vol 724. Springer, Singapore. https://doi.org/10.1007/978-981-33-4069-5_39
Download citation
DOI: https://doi.org/10.1007/978-981-33-4069-5_39
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-33-4068-8
Online ISBN: 978-981-33-4069-5
eBook Packages: Computer ScienceComputer Science (R0)