An optimal B-spline collocation technique for numerical simulation of viscous coupled Burgers’ equation

., Shallu; Kukreja, Vijay Kumar

doi:10.22034/cmde.2021.46178.1936

An optimal B-spline collocation technique for numerical simulation of viscous coupled Burgers’ equation

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Sant Longowal Institute of Engineering and Technology‎, ‎Longowal‎, ‎Punjab‎, ‎India.

10.22034/cmde.2021.46178.1936

Abstract

In this paper, an optimal cubic B-spline collocation method is applied to solve the viscous coupled Burgers’ equation, which helps in modeling the polydispersive sedimentation. As it is not possible to obtain optimal order of convergence with the standard collocation method, so to overcome this, posteriori corrections are made in cubic B-spline interpolant and its higher-order derivatives. This optimal cubic B-spline collocation method is used for space integration and for time-domain integration, the Crank-Nicolson scheme is applied along with the quasilinearization process to deal with the nonlinear terms in the equations. Von-Neumann stability analysis is carried out to discuss the stability of the technique. Few test problems are solved numerically along with the calculation of L2, L∞ error norms as well as the order of convergence. The obtained results are compared with those available in the literature, which shows the improvement in results over the standard collocation method and many other existing techniques also.

Keywords

20.1001.1.23453982.2022.10.4.13.2

References

[1] R. Abazari and A. Borhanifar, Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Comput. Math. Appl., 59(8) (2010), 2711–2722.
[2] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math., 181(2) (2005), 245–251.
[3] K. K. Ali, K. R. Raslan, and T. S. El-Danaf, Non-polynomial spline method for solving coupled Burgers equations, Comput. Methods Differ. Equ., 3(3) (2015), 218–230.
[4] E. Ashpazzadeh, B. Han, and M. Lakestani, Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation, J. Comput. Appl. Math., 317 (2017), 510–534.
[5] A. Bashan, A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number, Physica A, 545 (2020), 123755.
[6] R. E. Bellman and R. E. Kalaba, Quasilinearization and nonlinear boundary-value problems, New York : Elsevier, 1965.
[7] H. P. Bhatt and A. Q. M. Khaliq, Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Commun., 200 (2016), 117–138.
[8] N. Chuathong and S. Kaennakham, Numerical solution to coupled Burgers’ equations by Gaussian-based Hermite collocation scheme, J. Appl. Math., 2018.
[9] J. W. Daniel and B. K. Swartz, Extrapolated collocation for two-point boundary-value problems using cubic splines, IMA J. Appl. Math., 16(2) (1975), 161–174.
[10] M. Dehghan, B. N. Saray, and M. Lakestani, Mixed finite difference and Galerkin methods for solving Burgers equations using interpolating scaling functions, Math. Methods Appl. Sci., 37(6) (2014), 894–912.
[11] O. Ersoy and I. Dag, A trigonometric cubic B-spline finite element method for solving the nonlinear coupled Burger equation, arXiv preprint arXiv:1604.04419, 2016.
[12] S. E. Esipov, Coupled Burgers equations: a model of polydispersive sedimentation, Phys. Rev. E, 52(4) (1995), 3711.
[13] N. Fisher and B. Bialecki, Extrapolated ADI Crank–Nicolson orthogonal spline collocation for coupled Burgers’ equations, J. Differ. Equ. Appl., 26(1) (2020), 45–73.
[14] H. E. Gadain, Solving coupled pseudo-parabolic equation using a modified double Laplace decomposition method, Acta Math. Sci., 38(1) (2018), 333–346.
[15] M. Ghasemi, An efficient algorithm based on extrapolation for the solution of nonlinear parabolic equations, Int. J. Nonlin. Sci. Num., 19(1) (2018), 37–51.
[16] A. Jafarabadi and E. Shivanian, Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method, Eng. Anal. Bound. Elem., 95 (2018), 187–199.
[17] D. Kaya, An explicit solution of coupled viscous Burgers’ equation by the decomposition method, Int. J. Math. Math. Sci., 27(11) (2001), 675–680.
[18] A. H. Khater, R. S. Temsah, and M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math., 222(2) (2008), 333–350.
[19] M. Kumar and S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Commun., 185(3) (2014), 809–817.
[20] S. Kutluay and Y. Ucar, Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method, Math. Methods Appl. Sci., 36(17) (2013), 2403–2415.
[21] T. R. Lucas, Error bounds for interpolating cubic splines under various end conditions, SIAM J. Numer. Anal., 11(3) (1974), 569–584.
[22] R. C. Mittal and G. Arora, Numerical solution of the coupled viscous Burgers’ equation, Commun. Nonlinear. Sci., 16(3) (2011), 1304–1313.
[23] R. C. Mittal and R. Jiwari, Differential quadrature method for numerical solution of coupled viscous Burgers’ equations, Int. J. Comput. Methods Eng. Sci. Mech., 13(2) (2012), 88–92.
[24] R. C. Mittal and R. Rohila, A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations, Wave motion, 80 (2018), 47–68.
[25] R. C. Mittal and A. Tripathi, A collocation method for numerical solutions of coupled Burgers’ equations, Int. J. Comput. Methods Eng. Sci. Mech., 15(5) (2014), 457–471.
[26] J. Nee and J. Duan, Limit set of trajectories of the coupled viscous Burgers’ equations, Appl. Math. Lett., 1(1) (1998), 57–61.
[27] P. M. Prenter, Splines and Variational Methods, New York : Wiley-interscience publication, 1975.
[28] A. Rashid and A. I. B. M. Ismail, A Fourier pseudospectral method for solving coupled viscous Burgers equations, Comput. Methods. Appl. Math., 9(4) (2009), 412–420.
[29] K. R. Raslan, T. S. El-Danaf, and K. K. Ali, Collocation method with cubic trigonometric B-spline algorithm for solving coupled Burgers’ equation, Far East J. Appl. Math., 95(2) (2016), 109.
[30] R. Rohila and R. C. Mittal, Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method, Math. Sci., 12(2) (2018), 79–89.
[31] P. Roul, A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid, Appl. Numer. Math., 153 (2020), 558–574.
[32] P. Roul, A fourth order numerical method based on B-spline functions for pricing Asian options, Comput. Math. Appl., 80(3) (2020), 504–521.
[33] P. Roul and V. P. Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems, Appl. Math. Comput., 341 (2019), 428–450.
[34] P. Roul and V. P. Goura, A high-order B-spline collocation scheme for solving a nonhomogeneous time-fractional diffusion equation, Math. Methods Appl. Sci., 44(1) (2021), 546–567.
[35] B. N. Saray, M. Lakestani, and M. Dehghan, On the sparse multiscale representation of 2-D Burgers equations by an efficient algorithm based on multiwavelets, Numer. Methods Partial Differ. Equ., (2021).
[36] M. A. Shallal, K. K. Ali, K. R. Raslan, and A. H. Taqi, Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations, Arab. J. Basic Appl. Sci., 26(1) (2019), 331–341.
[37] Shallu and V. K. Kukreja, Analysis of RLW and MRLW equation using an improvised collocation technique with SSP-RK43 scheme, Wave Motion, (2021), p.102761.
[38] Shallu and V. K. Kukreja, An improvised collocation algorithm with specific end conditions for solving modified Burgers equation, Numer. Methods Partial Differ. Equ., 37(1) (2021), 874–896.
[39] Shallu, A. Kumari, and V. K. Kukreja, An efficient superconvergent spline collocation algorithm for solving fourth order singularly perturbed problems, Int. J. Appl. Comput. Math., 6(5) (2020), 1–23.
[40] Shallu, A. Kumari, and V. K. Kukreja, An improved extrapolated collocation technique for singularly perturbed problems using cubic B-spline Functions, Mediterr. J. Math., 18(4) (2021), 1–29.
[41] A. A. Soliman, The modified extended tanh-function method for solving Burgers-type equations, Physica A, 361(2) (2006), 394–404.
[42] V. K. Srivastava, M. K. Awasthi, and M. Tamsir, A fully implicit finite-difference solution to one dimensional coupled nonlinear Burgers’ equations, Int. J. Math. Math. Sci., 7(4) (2013), 23.
[43] H. Zadvan and J. Rashidinia, Development of non polynomial spline and new B-spline with application to solution of Klein-Gordon equation, Comput. Methods Differ. Equ., 8(4) (2020), 794–814.