Skip to main content
SpringerLink
Log in

An implicit finite difference scheme for the numerical solutions of two-dimensional Burgers equations

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this study, the system of two-dimensional Burgers equations is solved by a new approximation that approaches the solution at two time legs: approximation is explicit in x-direction and implicit in y-direction at the first leg while approximation is implicit in x-direction and explicit in y-direction at the second leg. Two test problems are used to illustrate the accuracy of the present approximation. Comparisons are made with the existing methods in the literature. The approximation is analyzed by von-Neumann stability analysis method and it is displayed that the approximation is unconditionally stable. The method is shown to be consistent and second order accurate in time and space. The obtained results show that the present approximation is successful to solve the system of two-dimensional Burgers equations.

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
Fig. 6

Similar content being viewed by others

References

  1. M. A. Abdou, A. A. Soliman, Variational iteration method for solving Burgers’ and coupled Burgers’ equations, Journal of Computational and Applied Mathematics, 181 (2005), 245–251.

    Article  MathSciNet  Google Scholar 

  2. G. Adomian, The diffusion - Brusselator equation, Computational Math. Appl., 29 (1995), 1–3.

    Article  MathSciNet  Google Scholar 

  3. H. Ahmad, T. A. Khan, C. Cesarano, Numerical Solutions of Coupled Burgers’ Equations, Axioms, 8 (2019), 1–17.

    Article  Google Scholar 

  4. A. R. Bahadir, A fully implicit finite-difference scheme for two dimensional Burgers’ equation, Applied Mathematics and Computation, 137(2003), 131–137.

    Article  MathSciNet  Google Scholar 

  5. J. M. Burger, A Mathematical model illustrating the theory of Turbulence, Adv. Appl. Mech., 1 (1948), 171–199.

    Article  MathSciNet  Google Scholar 

  6. G. Celikten, E. N. Aksan, Alternating Direction Implicit (ADI) Method for Numerical Solutions of 2-D Burgers Equations, Thermal Science, 23(1) (2019), S243–S252.

    Article  Google Scholar 

  7. J. D. Cole, On a Quasilinear parabolic equations occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225–236.

    Article  MathSciNet  Google Scholar 

  8. S. M. El Sayed, D. Kaya, On the numerical solution of the system of two-dimensional Burgers’ equations by the decomposition method, Applied Mathematics and Computation, 158 (2004), 101–109.

    Article  MathSciNet  Google Scholar 

  9. C. A. J. Fletcher, Generating exact solutions of the two dimensional Burgers’ equation, International Journal for Numerical Methods in Fluids, 3 (1983), 213–216.

    Article  Google Scholar 

  10. C. A. J. Fletcher, A comparison of finite element and finite difference of the one- and two-dimensional Burgers’ equations, Journal of Computational Physics, 51 (1983), 159–188.

    Article  MathSciNet  Google Scholar 

  11. O. Goyon, Multilevel schemes for solving unsteady equations, International Journal for Numerical Methods in Fluids, 22 (1996), 937–959.

    Article  MathSciNet  Google Scholar 

  12. P. C. Jain, D. N. Holla, Numerical solution of coupled Burgers equations. International Journal for Numerical Methods in Engineering, 12 (1978), 213–222.

    MATH  Google Scholar 

  13. F. Liu, S. Weiping, Numerical solutions of two-dimensional Burgers’ equations by lattice Boltzmann method, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 150–157.

    Article  MathSciNet  Google Scholar 

  14. R. C. Mittal, R. Jiwari, Differential quadrature method for two-dimensional Burgers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10(2009), 450–459.

    Article  MathSciNet  Google Scholar 

  15. H. S. Shukla, M. Tamsir, V. K. Srivastava, J. Kumar, Numerical Solution of two dimensional coupled viscous Burgers’ Equation using the Modified Cubic B-Spline Differential Quadrature Method, AIP Advances, 4 (117134)(2014), 1–10.

    Google Scholar 

  16. V. K. Srivastava, M. Tamsir, Y. Bhardwaj, Y. V. S. S. Sanyasiraju, Crank–Nicolson scheme for numerical solutions of two dimensional coupled Burgers’ equations, International Journal of Scientific and Engineering Research, 2 (5) (2011), 1–7.

    Google Scholar 

  17. V. K. Srivastava, S. Singh, M. K. Awasthi, Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme, AIP Advances, 3(082131)(2013), 1–7.

    Google Scholar 

  18. V. K. Srivastava, M. K. Awasthi, S. Singh, An implicit logarithm finite difference technique for two dimensional coupled viscous Burgers’ equation, AIP Advances, 3(122105) (2013), 1–9.

    Google Scholar 

  19. S. S. Wani, S. Thakar, Linear method for two dimensional Burgers’ equation, Ultra Scientist, 25 (1A) (2013), 156–168.

    Google Scholar 

  20. F. W. Wubs, E. D. de Goede, An explicit–implicit method for a class of time-dependent partial differential equations, Applied Numerical Mathematics, 9 (1992), 157–181.

    Article  MathSciNet  Google Scholar 

  21. G. Zhao, X. Yu, R. Zhang, The new numerical method for solving the system of two-dimensional Burgers’ equations, Computers and Mathematics with Applications, 62 (2011), 3279–3291.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100, Kars, Turkey

    Gonca Çelikten

Authors
  1. Gonca Çelikten
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gonca Çelikten.

Additional information

Communicated by NM Bujurke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Çelikten, G. An implicit finite difference scheme for the numerical solutions of two-dimensional Burgers equations. Indian J Pure Appl Math 53, 246–260 (2022). https://doi.org/10.1007/s13226-021-00031-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-021-00031-w

Keywords

Mathematics Subject Classification

Search

Navigation