Current issue
Archive
About the Journal
About
Editorial Board
Abstract & Indexing
Contact
Editorial Policies
Authorship & COI
Ethical principles
Principles of Transparent Checklist
Open access
For Authors
Instructions to Authors
Detailing Guidelines for Authors
Review process
Review sheet
How to submit
Open Access license
ORIGINAL PAPER
A Numerical Approximation of 2D Coupled Burgers’ Equation Using Modified Cubic Trigonometric B-Spline Differential Quadrature Method
1
Department of Mathematics, Lovely Professional University, Punjab, 144411, India
Online publication date: 2022-08-29
Publication date: 2022-09-01
International Journal of Applied Mechanics and Engineering 2022;27(3):79-102
KEYWORDS
trigonometric B-splinedifferential quadrature method (DQM)strong stability preserving Runge-Kutta 43 (SSP-RK43) schemeL2 and L∞ error normscoupled 2D non-linear Burgers’ equation
ABSTRACT
In the present paper, trigonometric B-spline DQM is applied to get the approximated solution of coupled 2D non-linear Burgers’ equation. This technique, named modified cubic trigonometric B-spline DQM, has been used to obtain accurate and effective numerical approximations of the above-mentioned partial differential equation. For checking the compatibility of results, different types of test examples are discussed. A comparison is done between L2 and L∞ error norms with the previous, present results and with the exact solution. The resultant set of ODEs has been solved by employing the SSP RK 43 method. It is observed that the obtained results are improved compared to the previous numerical results in the literature.
REFERENCES (38)
1.
Fletcher C.A.J. (1983): Generating exact solutions of the two-dimensional Burgers’ equation.– Int. J. Numer. Meth. Fluids, vol.3, pp.213-216.
2.
Jain P.C. and Holla D.N. (1978): Numerical solution of coupled Burgers’ equations.– Int. J. of Non-Linear Mech., vol.13, pp.213-222.
3.
Fletcher A.J. (1983): A comparison of finite element and finite difference of the one- and two dimensional Burgers’ equations.– J. Comput. Phys., vol.51, pp.159-188.
4.
Wubs F.W. and de Goede E.D. (1992): An explicit–implicit method for a class of time-dependent partial differential equations.– Appl. Numer. Math., vol.9, pp.157-181.
5.
Goyon O. (1996): Multilevel schemes for solving unsteady equations.– Int. J. Numer. Meth. Fluids, vol.22, pp.937-959.
6.
Bahadir A.R. (2003): A fully implicit finite-difference scheme for two-dimensional Burgers’ equation.– Applied Mathematics and Computation, vol.137, pp.131-137.
7.
Srivastava V.K., Tamsir M., Bhardwaj U. and Sanyasiraju Y. (2011): Crank-Nicolson scheme for numerical solutions of two dimensional coupled Burgers’ equations.– IJSER, vol.2, No.5, p.44.
8.
Tamsir M. and Srivastava V.K. (2011): A semi-implicit finite-difference approach for two- dimensional coupled Burgers’ equations.– IJSER, vol.2, No.6, p.46.
9.
Srivastava V.K. and Tamsir M. (2012): Crank-Nicolson semi-implicit approach for numerical solutions of two-dimensional coupled nonlinear Burgers’ equations.– Int. J. Appl. Mech. Eng., vol.17, No.2, pp.571-581.
10.
Srivastava V.K., Awasthi M.K. and Tamsir M. (2013): A fully implicit Finite-difference solution to one-dimensional Coupled Nonlinear Burgers’ equations.– Int. J. Math. Sci., vol.7, No.4, p.23.
11.
Srivastava V.K., Awasthi M.K. and Singh S. (2013): An implicit logarithmic finite difference technique for two dimensional coupled viscous Burgers’ equation.– AIP Advances, vol.3, Article ID.122105, p.9.
12.
Srivastava V.K., Singh S. and Awasthi M.K. (2013): Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme.– AIP Advances, vol.3, Article ID.082131, p.7.
13.
Cole J.D. (1951): On a quasilinear parabolic equations occurring in aerodynamics.– Q. Appl. Math., vol.9, pp.225-236.
14.
Fletcher C.A.J (1983): A comparison of finite element and finite difference of the one and two dimensional Burgers’ equations.– J. Comput. Phys. vol.51, pp.159-188.
15.
Kutluay S., Bahadir A.R. and Ozdes A. (1999): Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods.– J. Comput. Appl. Math., vol.103, pp.251-261.
16.
Liao W (2008): An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation.– Appl. Math. Comput., vol.206, pp.755-764.
17.
Ozis T., Esen A. and Kutluay S. (2005): Numerical solution of Burgers’ equation by quadratic B-spline finite elements.– Appl. Math. Comput., vol.165, pp.237-249.
18.
Hassanien I.A., Salama A.A., Hosham H.A. (2005): Fourth-order finite difference method for solving Burgers’ equation.– Appl. Math. Comput., vol.170, pp.781-800.
19.
Dag I., Irk D. and Sahin A. (2005): B-Spline collocation methods for numerical solutions of the Burgers’ equation.– Math. Probl. Eng., vol.5, pp.521-538.
20.
Korkmaz A. and Dag I. (2011): Shock wave simulations using sinc differential quadrature method.– Eng. Comput. Int. J. Comput. Aided Eng. Softw., vol.28, No.1, pp.654-674.
21.
Korkmaz A. and Dag I. (2011): Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation.– J. Frankl. Inst., vol.348, No.10, 2863-2875.
22.
Korkmaz A., Aksoy A.M. and Dag I. (2011): Quartic B-spline differential quadrature method.– Int. J. Nonlinear Sci., vol.11, No.4, pp.403-411.
23.
Mittal R.C. and Jain R.K. (2012): Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method.– Appl. Math. Comput., vol.218, pp.7839-7855.
24.
Korkmaz A, and Dag I. (2012): Cubic B-spline differential quadrature methods for the advection-diffusion equation.– Int. J. Numer. Methods Heat Fluid Flow, vol.22, No.8, pp.1021-1036.
25.
Arora G. and Singh B.K. (2013): Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method.– Appl. Math. Comput., vol.224, No.1, pp.166-177.
26.
Shukla H.S., Tamsir M., Srivastava V.K. and Kumar J. (2014): Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method.– AIP Adv., vol.4, Article ID.117134. p.10.
27.
Tamsir M., Srivastava V.K. and Jiwari R. (2016): An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation.– App. Math. and Comp., vol.290, pp.111-124.
28.
Mittal R.C. and Jiwari R. (2009): Differential quadrature method for two-dimensional Burgers’ equations.– Int. J. for Comp. Meth. in Eng. Sci. and Mech., vol.10, No.6, pp.450-459.
29.
Zhu H., Shu H. and Ding M. (2010): Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method.– Computers & Mathematics with Applications, vol.60, No.3, pp.840-848.
30.
Aminikhah H. (2012): A new efficient method for solving two-dimensional Burgers’ equation.– ISRN Computational Mathematics, vol.2012, https://doi.org/10.5402/2012/6....
31.
Biazar J. and Aminikhah H. (2009): Exact and numerical solutions for non-linear Burger’s equation by VIM.– Mathematical and Computer Modelling, vol.49, No.7-8, pp.1394-1400.
32.
Bert C.W. and Malik M. (1996): Differential quadrature method in computational mechanics: a review.– App. Mech. Rev., vol.49, No.1, p.28.
33.
Shu C. (2012): Differential Quadrature and its Application in Engineering.– Springer Science & Business Media, p.356.
34.
Bellman R., Kashef B.G. and Casti J. (1972): Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations.– J. Comput. Phys., vol.1, pp.40-52.
35.
Quan J.R. and Chang C.T. (1989): New insights in solving distributed system equations by the quadrature methods-I.– Comput. Chem. Eng., vol.13, pp.779-788.
36.
Quan J. R. and Chang C. T. (1989): New insights in solving distributed system equations by the quadrature methods-II.– Comput. Chem. Eng., vol.13, pp.1017–1024.
37.
Shu C. and Richards B.E. (1990): High resolution of natural convection in a square cavity by generalized differential quadrature.– in: Proceed. of third Conf. on Adv. Numer. Meth. Eng. Theory Appl., Swansea UK 2, pp.978-985.
38.
Arora G. and Singh B.K. (2013): Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method.– Applied Mathematics and Computation, vol.224, pp.166-177.
| eISSN: | 2353-9003 |
| ISSN: | 1734-4492 |
Task: edition of a scientific journal, its internationalization and ensuring open access to the journal via the Internet
© 2006-2024 Journal hosting platform by Bentus
We process personal data collected when visiting the website. The function of obtaining information about users and their behavior is carried out by voluntarily entered information in forms and saving cookies in end devices. Data, including cookies, are used to provide services, improve the user experience and to analyze the traffic in accordance with the Privacy policy. Data are also collected and processed by Google Analytics tool (more).
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
