Higher Order Splitting Method for Numerical Solution Of Nonlinear Coupled System Viscous Burger's Equation

Abstract

In this study, the numerical solutions of nonlinear coupled system viscous Burgers equation with appropriate initial and boundary conditions are going to be obtained by Strang splitting method and also Ext4 and Ext6 methods obtained by extrapolation technique. To apply splitting methods, coupled system viscous Burgers equation split up into two subequation, one is linear and the other is nonlinear equation. Cubic B-spline functions and derivatives are used for the dependent variables u(x,t) and v(x,t) in each sub-equation obtained. Numerical schemas were obtained by applying each sub-equation of the collocation finite element method and the stability analyzes were investigated by the von-Neumann method. The effectiveness of the method was tested on three commonly used test problems in the literature. It was observed that the calculated numerical results were in agreement with the exact solution and compared with the previous studies.

Keywords

• Operator Splitting method
• Coupled viscous Burgers equation
• Cubic B-spline functions
• Collocation method
• Strang splitting

References

1. [1] S.E. Esipov, Copled Burgers’ equations: a model of polydispersive sedimentation, Phys. Rev. E 52 (1995) 3711–3718.
2. [2] J. Nee, J. Duan, Limit set of trajectories of the coupled viscous Burgers’ equations. Appl. Math. Lett. 11(1) (1998) 57–61.
3. [3] T. A. Abassy, M. A. El-Tawil, H. El-Zoheiry, Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Pad´e technique, Computers and Mathematics with Applications 54 (2007) 940–954.
4. [4] P. C. Jain, M. K. kadalbajoo, Invariant Embedding Method for the Solution of Coupled Burgers’ Equations, Journal of Mathematical Analysis and Applications 72 (1979) l-16.
5. [5] M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burgers’ equations using Adomian–Pade technique, Applied Mathematics and Computation 189 (2007) 1034–1047.
6. [6] R.C. Mittal, G. Arora, Numerical solution of the coupled viscous Burgers’ equation, Commun Nonlinear Sci Numer Simulat 16 (2011) 1304–1313.
7. [7] R. C. Mittal and A. Tripathi, A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations, International Journal for Computational Methods in Engineering Science and Mechanics, 15 (2014) 457–471.
8. [8] S. ul-Islam, B. Sarler, R. Vertnik, G. Kosec, Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations, Applied Mathematical Modelling 36 (2012) 1148–1160.

9. [9] A. Rashid, M. Abbas, A. I. Md. Ismail, A. Abd Majid, Numerical solution of the coupled viscous Burgers equations by Chebyshev–Legendre Pseudo-Spectral method, Applied Mathematics and Computation 245 (2014) 372–381.
10. [10] A. Rashid and A.I.B. Ismail, A Fourier pseudospectral method for solving coupled viscous Burgers equations, Comput. Methods Appl. Math. 9 (2009) 412-420.
11. [11] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math. 222 (2008) 333-350.
12. [12] S. Kutluay and Y. Ucar, Numerical solutions of the coupled Burgers equation by the Galerkin quadratic B-spline finite element method, Math. Meth. Appl. Sci., 36 (2013) 2403–2415.
13. [13] Y. Ucar, Numerical Solutions of Coupled Differential Equations With B-Spline Finite Element Method, Ph.D. Thesis, ˙In¨on¨u University, 2011.
14. [14] V. K. Srivastava, M. Tamsir, M.K. Awasthi, S. Singh, One-dimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method, AIP Advances, 4 (2014) 037119-10.
15. [15] Q. Li, Z. Chai, B. Shi, A novel lattice Boltzmann model for the coupled viscous Burgers’ equations, Applied Mathematics and Computation 250 (2015) 948–957.
16. [16] H. Lai, C. Ma, A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation, Physica A 395 (2014) 445–457.
17. [17] R. Mokhtari, A. Samadi Toodar, N.G. Chegini, Application of the Generalized Differential Quadrature Method in Solving Burgers’ Equations, Commun. Theor. Phys. 56(6) (2011) 1009–1015.
18. [18] R. C. Mittal, R. Jiwari, Differential Quadrature Method for Numerical Solution of Coupled Viscous Burgers’ Equations, International Journal for Computational Methods in Engineering Science and Mechanics, 13 (2012) 88–92.
19. [19] D. Kaya, An Explicit Solution of Coupled Burgers’ Equations by Decomposition Method, IJMMS, 27 (2001) 3711–3718.
20. [20] A.A. Soliman, The modified extended tanh-function method for solving Burgers-type equations, Physica A 361 (2006) 394–404.
21. [21] R. Abazari, A. Borhanifar, Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Computers and Mathematics with Applications 59 (2010) 2711–2722.
22. [22] A. Başhan, A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number, Physica A 545 (2020) 123755
23. [23] Y. Uçar, N.M. Yağmurlu, A.Başhan, Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-spline Differential Quadrature Methods, Sigma J Eng & Nat Sci 37 (1), 2019, 129-142
24. [24] A. Başhan, S.B.G. Karakoc¸, T.Geyikli, B-spline Differential Quadrature Method for the Modified Burgers’ Equation, Çankaya University Journal of Science and Engineering Volume 12, No. 1 (2015) 001–013
25. [25] S.B.G. Karakoc¸, A. Bas¸han, T.Geyikli, Two Different Methods for Numerical Solution of the Modified Burgers’ Equation, The Scientific World Journal Volume 2014, Article ID 780269, 13 pages http://dx.doi.org/10.1155/2014/780269
26. [26] E. Bahar, G. Gurarslan, Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method, International Journal of Engineering & Applied Sciences (IJEAS), 9 (2017) 76-88.
27. [27] H. Holden, C. Lubich, N. H. Risebro, Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp. 82 (2013) 173-185.
28. [28] H. Holden, K. H. Karlsen, N. H. Risebro, Operator Splitting Methods for Generalized Korteweg–De Vries Equations, Journal of Computational Physics 153 (1999) 203–222.
29. [29] H. Wang, Numerical studies on the split-step finite difference method for nonlinear Schr¨odinger equations, Applied Mathematics and Computation 170 (2005) 17–35.
30. [30] X. Xiao, D. Gui, X. Feng, A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen–Cahn equation, International Journal of Numerical Methods for Heat & Fluid Flow, 27 (2) (2017) 530-542.
31. [31] J. Geiser, Iterative Splitting Methods for Differential Equations, Chapman & Hall/CRC 2011.
32. [32] B. Sportisse, An Analysis of Operator Splitting Techniques in the Stiff Case, Journal of Computational Physics 161 (2000) 140–168.
33. [33] G. Strang, SIAM J. NUMER. ANAL. On The Construction And Comparison Of Difference Schemes, 5(3) (1968) 506-517.
34. [34] S. MacNamara, G. Strang, Operator Splitting. In: Glowinski R., Osher S., Yin W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham, 2016.
35. [35] M. Seydaoğlu, U. Erdoğan, T. Öziş, Numerical solution of Burgers’ equation with high order splitting methods, Journal of Computational and Applied Mathematics, vol. 291 (2016) pp. 410–421. http://dx.doi.org/10.1016/j.cam.2015.04.021
36. [36] P.M. Prenter, Splines and Variatinoal Methods, John Wiley, New York (1975).
37. [37] S.G. Rubin , R.A. Graves, Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436. Washington, DC; 1975.
38. [38] J. VonNeumann and R. D. Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, J. Appl. Phys. 21 (1950) 232-237.