Skip to main content
SpringerLink
Log in

A Higher Order Numerical Implicit Method for Non-Linear Burgers’ Equation

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper proposes a higher order implicit numerical scheme to approximate the solution of the nonlinear partial differential equation (PDE). This equation is a simplified form of Navier–Stoke’s equation also known as Burgers’ equation. It is an important nonlinear PDE which arises frequently in mathematical modeling of turbulence in fluid dynamics. In order to handle nonlinearity a nonlinear transformation is used which converts the nonlinear PDE into a linear PDE. The linear PDE is semi-discretized in space by method of lines to yield a system of ordinary differential equations in time. The resulting system of differential equations is investigated and found to be a stiff system. A system of stiff differential equations is further discretized by a low-dispersion and low-dissipation implicit Runge–Kutta method and solved by using MATLAB 8.0. The proposed scheme is unconditionally stable. Moreover it is simple, easy to implement and requires less computational time. Finally, the adaptability of the scheme is demonstrated by means of numerical computations by taking three test problems. The present implicit scheme have been compared with existing schemes in literature which shows that the proposed scheme offers more accuracy with less computational time than the numerical schemes given in Jiwari (Comput Phys Comm 183:2413–2423, 2012), Kutluay et al. (J Comput Appl Math 103:251–261, 1998), Kutluay et al. (J Comput Appl Math 167:21–33, 2004).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Arora, G., Singh, B.K.: Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method. Appl. Math. Comput. 224, 166–177 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Asaithambi, A.: Numerical solution of the Burgers equation by automatic differentiation. Appl. Math. Comput. 216, 2700–2708 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Bateman, H.: Some recent researches in motion of fluids. Mon. Weather Rev. 43, 163–170 (1915)

    Article  Google Scholar 

  4. Benton, E.R., Platzman, G.W.: A table of solutions of the one-dimensional Burgers’ Equation. Q. Appl. Math. 30, 195–212 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhatti, M.I., Bhatta, D.D.: Numerical solution of Burgers’ equation in a B-polynomial basis. Phys. Scr. 73, 539–544 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burgers, J.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948)

    Article  MathSciNet  Google Scholar 

  7. Burgers, J.M.: Mathematical examples illustrating relation occurring in the theory of turbulent fluid motion. Trans. R. Neth. Acad. Sci. Amst. 17, 1–53 (1939)

    MathSciNet  MATH  Google Scholar 

  8. Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)

    Book  MATH  Google Scholar 

  9. Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 30, 225–236 (1972)

    Google Scholar 

  10. Dhawan, S., Kapoor, S., Kumar, S., Rawat, S.: Contemporary review of techniques for the solution of nonlinear Burgers’ equation. J. Comput. Sci. 3, 405–419 (2012)

    Article  Google Scholar 

  11. Evans, D.J., Abdullah, A.R.: The group explicit method for the solution of Burgers’ equation. Computing 32, 239–253 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Evans, G., Blackledge, J., Yardley, P.: Numerical Methods for Partial Differential Equations. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  13. Ganaie, I.A., Kukreja, V.K.: Numerical solution of Burgers’ equation by cubic Hermite collocation method. Appl. Math. Comput. 237, 571–581 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Gao, Y., Le, L.-H., Shi, B.-C.: Numerical solution of Burgers’ equation by lattice Boltzmann method. Appl. Math. Comput. 219, 7685–7692 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Guo, Y., Shi, Y., Li, Y.: A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation. Appl. Math. Comput. 181, 172–185 (2016)

    MathSciNet  Google Scholar 

  16. Haq, S., Hussain, A., Uddin, M.: On the numerical solution of nonlinear Burgers’ type equations using meshless method of lines. Appl. Math. Comput. 218, 6280–6290 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Hopf, E.: The partial differential equation \(u_t+uu_x=\nu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230 (1950)

    Article  MathSciNet  Google Scholar 

  18. Jiwari, R.: A haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput. Phys. Comm. 183, 2413–2423 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kadalbajoo, M.K., Awasthi, A.: A numerical method based on Crank–Nicolson scheme for Burgers’ equation. Appl. Math. Comput. 182, 1430–1442 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Kadalbajoo, M.K., Sharma, K.K., Awasthi, A.: A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equation. Appl. Math. Comput. 170, 1365–1393 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Kutluay, S., Bahadir, A.R., Ozdes, A.: Numerical solution of one-dimensional Burgers’ equation: explicit and exact explicit methods. J. Comput. Appl. Math. 103, 251–261 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kutluay, S., Esen, A., Dag, I.: Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math. 167, 21–33 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Prakash, A., Kumar, M., Sharma, K.K.: Numerical method for solving fractional coupled Burgers equations. Appl. Math. Comput. 262, 314–320 (2015)

    MathSciNet  Google Scholar 

  24. Mukundan, V., Awasthi, A.: A comparative study of three level explicit and implicit numerical scheme for convection diffusion equation. In: Proceedings of International conference on Mathematical and computational sciences, pp. 58–64. Narosa Publishing house (2015)

  25. Mukundan, V., Awasthi, A.: Efficient numerical techniques for Burgers’ equation. Appl. Math. Comput. 262, 282–297 (2015)

    MathSciNet  Google Scholar 

  26. Rashidi, M.M., Ganji, D.D., Dinarvand, S.: Explicit analytical solutions of the generalized Burger and Burger–Fisher Equations by homotopy perturbation method. Numer. Methods Partial. Differ. Equ. 25(2), 409–417 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rashidi, M.M., Domairry, G., Dinarvand, S.: Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14(3), 708–717 (2009)

    Article  Google Scholar 

  28. Rashidi, M.M., Erfani, E.: New analytical method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM. Comput. Phys. Commun. 180, 1539–1544 (2009)

    Article  MathSciNet  Google Scholar 

  29. Rothe, E.: Zweidimesionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math. Ann. 102, 650–670 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  30. Najafi-Yazdi, A., Mongeau, L.: A low-dispersion and low-dissipation implicit Runge–Kutta scheme. J. Comput. Phys. 233, 315–323 (2013)

    Article  MathSciNet  Google Scholar 

  31. Ozin, T., Aksan, E.N., Ozdes, A.: A finite element approach for solution of Burgers’ equation. Appl. Math. Comput. 139, 417–428 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Shao, L., Feng, X., He, Y.: The local discontinuous Galerkin finite element method for Burgers’ equation. Math. Comput. Model. 54, 2943–2954 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shukla, H.S., Tamsir, M., Srivastava, V.K., Rashidi, M.M.: Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation. Mod. Phys. Lett. B (in press)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Calicut, Kozhikode, 673 601, Kerala, India

    Vijitha Mukundan & Ashish Awasthi

Authors
  1. Vijitha Mukundan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ashish Awasthi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashish Awasthi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukundan, V., Awasthi, A. A Higher Order Numerical Implicit Method for Non-Linear Burgers’ Equation. Differ Equ Dyn Syst 25, 169–186 (2017). https://doi.org/10.1007/s12591-016-0318-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-016-0318-6

Keywords

Mathematics Subject Classification

Search

Navigation