Efficient Chebyshev Pseudospectral Methods for Viscous Burgers’ Equations in One and Two Space Dimensions

  • Mahboub BaccouchEmail author
  • Slim Kaddeche
Original Paper


Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. In this paper, we propose an efficient and accurate numerical method for the one and two dimensional nonlinear viscous Burgers’ equations and coupled viscous Burgers’ equations with various values of viscosity subject to suitable initial and boundary conditions. The method is based on the Chebyshev collocation technique in space and the fourth-order Runge–Kutta method in time. This proposed scheme is robust, fast, flexible, and easy to implement using modern mathematical software such as Matlab. Furthermore, it can be easily modified to handle other problems. Extensive numerical simulations are presented to demonstrate the accuracy of the proposed scheme. The numerical solutions for different values of the Reynolds number are compared with analytical solutions as well as other numerical methods available in the literature. Compared to other numerical methods, the proposed method is shown to have higher accuracy with fewer nodes. Our numerical experiments show that the Chebyshev collocation method is an efficient and reliable scheme for solving Burgers’ equations with reasonably high Reynolds number.


Viscous Burgers’ equations Coupled viscous Burgers’ equations Chebyshev pseudospectral method Chebyshev collocation method Reynolds number 



  1. 1.
    Abdou, M., Soliman, A.: Variational iteration method for solving Burger’s and coupled Burger’s equations. J. Comput. Appl. Math. 181(2), 245–251 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aliyu, A., Inc, M., Yusuf, A., Baleanu, D.: Symmetry analysis, explicit solutions, and conservation laws of a sixth-order nonlinear ramani equation. Symmetry 10(8), 341 (2018)zbMATHGoogle Scholar
  3. 3.
    Asaithambi, A.: Numerical solution of the Burgers’ equation by automatic differentiation. Appl. Math. Comput. 216, 2700–2708 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ashyralyev, A., Gambo, Y.Y.: Modified Crank–Nicholson difference schemes for one dimensional nonlinear viscous Burgers’ equation for an incompressible flow. AIP Conf. Proc. 1759(1), 020100 (2016)Google Scholar
  5. 5.
    Aswin, V.S., Awasthi, A., Rashidi, M.M.: A differential quadrature based numerical method for highly accurate solutions of Burgers’ equation. Numer. Methods Partial Differ. Equ. 33(6), 2023–2042 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bahadir, A.: A fully implicit finite-difference scheme for two-dimensional Burgers’ equations. Appl. Math. Comput. 137(1), 131–137 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bakodah, H., Al-Zaid, N., Mirzazadeh, M., Zhou, Q.: Decomposition method for solving Burgers’ equation with dirichlet and neumann boundary conditions. Opt. Int. J. Light Electron Opt. 130(Supplement C), 1339–1346 (2017)Google Scholar
  8. 8.
    Bateman, H.: Some recent researches on the motion of fluids. Mon. Weather Rev. 43, 163–170 (1915)Google Scholar
  9. 9.
    Bhatt, H., Khaliq, A.: Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation. Comput. Phys. Commun. 200(Supplement C), 117–138 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bhrawy, A., Zaky, M., Baleanu, D.: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Romanian Rep. Phys. 67(2), 340–349 (2015)Google Scholar
  11. 11.
    Biazar, J., Aminikhah, H.: Exact and numerical solutions for non-linear Burger’s equation by vim. Math. Comput. Model. 49(7), 1394–1400 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Binous, H., Bellagi, A.: Orthogonal collocation methods using mathematica in the graduate chemical engineering curriculum. Comput. Appl. Eng. Educ. 24(1), 101–113 (2016)Google Scholar
  13. 13.
    Binous, H., Kaddeche, S., Abdennadher, A., Bellagi, A.: Numerical elucidation of three-dimensional problems in the chemical engineering graduate curriculum. Comput. Appl. Eng. Educ. 24(6), 866–875 (2016)Google Scholar
  14. 14.
    Binous, H., Kaddeche, S., Bellagi, A.: Solving two-dimensional chemical engineering problems using the Chebyshev orthogonal collocation technique. Comput. Appl. Eng. Educ. 24(1), 144–155 (2016)Google Scholar
  15. 15.
    Binous, H., Kaddeche, S., Bellagi, A.: Solution of six chemical engineering problems using the Chebyshev orthogonal collocation technique. Comput. Appl. Eng. Educ. 25(4), 594–607 (2017)Google Scholar
  16. 16.
    Binous, H., Shaikh, A.A., Bellagi, A.: Chebyshev orthogonal collocation technique to solve transport phenomena problems with matlab and mathematica. Comput. Appl. Eng. Educ. 23(3), 422–431 (2015)Google Scholar
  17. 17.
    Burgers, J.: A mathematical model illustrating the theory of turbulence, vol. 1. In: Advances in Applied Mechanics, Elsevier, pp. 171–199 (1948)Google Scholar
  18. 18.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)zbMATHGoogle Scholar
  19. 19.
    Dag, I., Canivar, A., Sahin, A.: Taylor Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2696–2708 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    David Gottlieb, S.A.O.: Numerical Analysis of Spectral Methods: Theory and Applications. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial Mathematics (1987)Google Scholar
  21. 21.
    Elgindy, K.T., Dahy, S.A.: High-order numerical solution of viscous Burgers’ equation using a Cole-Hopf barycentric gegenbauer integral pseudospectral method. Math. Methods Appl. Sci. 1–26 (2018)Google Scholar
  22. 22.
    Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  23. 23.
    Ganaie, I., Kukreja, V.: Numerical solution of Burgers’ equation by cubic hermite collocation method. Appl. Math. Comput. 237(Supplement C), 571–581 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Goyon, O.: Multilevel schemes for solving unsteady equations. Int. J. Numer. Methods Fluids 22(10), 937–959 (1996)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hu, X., Huang, P., Feng, X.: A new mixed finite element method based on the Crank–Nicolson scheme for Burgers’ equation. Appl. Math. 61(1), 27–45 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jiwari, R.: A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput. Phys. Commun. 188(Supplement C), 59–67 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Jiwari, R., Alshomrani, A.S.: A new algorithm based on modified trigonometric cubic B-splines functions for nonlinear Burgers’-type equations. Int. J. Numer. Methods Heat Fluid Flow 27(8), 1638–1661 (2017)Google Scholar
  28. 28.
    Johnston, S.J., Jafari, H., Moshokoa, S.P., Ariyan, V.M., Baleanu, D.: Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order. Open Phys. 85(2), 247–252 (2016)Google Scholar
  29. 29.
    Kaya, D.: An explicit solution of coupled viscous Burgers’ equation by the decomposition method. Bull. Malays. Math. Sci. Soc. 27(11), 675–680 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Khan, M.: A novel solution technique for two dimensional Burgers’ equation. Alex. Eng. J. 53(2), 485–490 (2014)Google Scholar
  31. 31.
    Khater, A., Temsah, R., Hassan, M.: A Chebyshev spectral collocation method for solving Burgers’-type equations. J. Comput. Appl. Math. 222(2), 333–350 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. 85(2), 699–715 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kutluay, S., Ucar, Y., Yagmurlu, N.M.: Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method. Bull. Malays. Math. Sci. Soc. 39(4), 1603–1614 (2016)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Liu, F., Wang, Y., Li, S.: Barycentric interpolation collocation method for solving the coupled viscous Burgers’ equations. Int. J. Comput. Math. 0(ja), 1–13 (2017)Google Scholar
  35. 35.
    Mittal, R., Jain, R.: Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218, 7839–7855 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Mittal, R., Jiwari, R.: A differential quadrature method for numerical solutions of Burgers’-type equations. Int. J. Numer. Methods Heat Fluid Flow 22(7), 880–895 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mohanty, R., Dai, W., Han, F.: Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers’ equations. Appl. Math. Comput. 256, 381–393 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Mukundan, V., Awasthi, A.: Efficient numerical techniques for Burgers’ equation. Appl. Math. Comput. 262(Supplement C), 282–297 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mukundan, V., Awasthi, A.: A higher order numerical implicit method for non-linear Burgers’ equation. Differ. Equ. Dyn. Syst. 25(2), 169–186 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Rahman, K., Helil, N., Yimin, R.: Some new semi-implicit finite difference schemes for numerical solution of Burgers equation. In: International Conference on Computer Application and System Modeling (ICCASM 2010)—IEEE 2010 International Conference on Computer Application and System Modeling (ICCASM 2010)—Taiyuan, China (2010.10.22-2010.10.24) (2010) V14–451–V14–455Google Scholar
  41. 41.
    Rashid, A., Ismail, A.I.B.M.: A Fourier pseudospectral method for solving coupled viscous Burgers equations. Comput. Methods Appl. Math. 9(4), 412–420 (2009)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Saad, K.M., Atangana, A., Baleanu, D.: New fractional derivatives with non-singular kernel applied to the Burgers equation. Chaos Interdiscip. J. Nonlinear Sci. 28(6), 063109 (2018)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Seydaoglu, M., Erdogan, U., Ozis, T.: Numerical solution of Burgers’ equation with high order splitting methods. J. Comput. Appl. Math. 291(Supplement C), 410–421 (2016). mathematical Modeling and Computational MethodsMathSciNetzbMATHGoogle Scholar
  44. 44.
    Shi, F., Zheng, H., Cao, Y., Li, J., Zhao, R.: A fast numerical method for solving coupled Burgers’ equations. Numer. Methods Partial Differ. Equ. 33(6), 1823–1838 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In: Bertoluzza, S., Falletta, S., Russo, G., Shu, C.-W. (eds.) Numerical Solutions of Partial Differential Equations. Advanced Courses in Mathematics, pp. 149–201. CRM, Barcelona (2009)Google Scholar
  46. 46.
    Shukla, H.S., Tamsir, M., Srivastava, V.K., Kumar, J.: Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method. AIP Adv. 4(11), 117134 (2014)Google Scholar
  47. 47.
    Singh, B.K., Kumar, P.: A novel approach for numerical computation of Burgers’ equation in (1+1) and (2+1) dimensions. Alex. Eng. J. 55(4), 3331–3344 (2016)Google Scholar
  48. 48.
    Srivastava, V., Awasthi, M., Tamsir, M.: A fully implicit finite-difference solution to one dimensional coupled nonlinear Burgers’ equations. Int. J. Math. Math. Sci. 7(4), 23–28 (2013)zbMATHGoogle Scholar
  49. 49.
    Srivastava, V.K., Singh, S., Awasthi, M.K.: Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme. AIP Adv. 3(8), 082131 (2013)Google Scholar
  50. 50.
    Tamsir, M., Srivastava, V.K., Jiwari, R.: An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation. Appl. Math. Comput. 290(Supplement C), 111–124 (2016)MathSciNetGoogle Scholar
  51. 51.
    Trefethen, L.N.: Spectral Methods in MatLab. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2000)Google Scholar
  52. 52.
    ul Islam, S., Sarler, B., Vertnik, R., Kosec, G.: Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations. Appl. Math. Model. 36(3), 1148–1160 (2012)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Venkatesh, S.G., Ayyaswamy, S.K., Raja Balachandar, S.: An approximation method for solving Burgers’ equation using legendre wavelets. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 87(2), 257–266 (2017)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Wazwaz, A.-M.: Partial Differential Equations: Methods and Applications. Taylor & Francis, Lisse (2002)zbMATHGoogle Scholar
  55. 55.
    Yousefi, M., Rashidinia, J., Yousefi, M., Moudi, M.: Numerical solution of Burgers’ equation by B-spline collocation. Afr. Mat. 27(7), 1287–1293 (2016)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Zhang, X.H., Ouyang, J., Zhang, L.: Element-free characteristic Galerkin method for Burgers’ equation. Eng. Anal. Bound. Elem. 33(3), 356–362 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska at OmahaOmahaUSA
  2. 2.Department of Physics, National Institute of Applied Sciences and TechnologyLaboratory Materials, Measurements and Applications (LR 11 ES 25)TunisTunisia

Personalised recommendations