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Weak Galerkin finite element method for solving one-dimensional coupled Burgers’ equations

  • Ahmed J. HusseinEmail author
  • Hashim A. Kashkool
Original Research

Abstract

In this paper, we apply a weak Galerkin method for solving one dimensional coupled Burgers’ equations. Based on a conservation form for nonlinear term and some of the technical derivational. Theorticly, we drive the optimal order error in \(L^2\) and \(H^1\) norm for both continuous and discrete time weak Galerkin finite element schemes, also the stability of continuous time weak Galerkin finite element method is proved. Numerically, the accuracy and effectiveness of the weak Galerkin finite element method are illustrated by using Numerical examples with the lower order Raviart–Thomas element \(RT_k\) for discrete weak derivative space.

Keywords

Weak Galerkin finite element method (WG-FEM) Burgers’ equations Optimal order 

Mathematics Subject Classification

65N15 65N30 

Notes

References

  1. 1.
    Zhang, R., Yu, X., Zhao, G.: Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations. Chin. Phys. B 20(11), 110205 (2011)CrossRefGoogle Scholar
  2. 2.
    Kaya, D.: An explicit solution of coupled viscous Burgers’ equations by the decomposition method. JJMMS 27(11), 675 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Soliman, A.A.: The modified extended tanh-function method for solving Burgers-type equations. Physica A 361, 394 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Esipov, S.E.: Coupled Burgers’ equations: a model of polydispersive sedimentation. Phys. Rev. E 52, 3711 (1995)CrossRefGoogle Scholar
  5. 5.
    Abdou, M.A., Soliman, A.A.: Variational iteration method for solving Burgers’ and coupled Burgers’ equations. J. Comput. Appl. Math. 181(2), 245–251 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wei, G.W., Gu, Y.: Conjugate filter approach for solving Burgers’ equation. J. Comput. Appl. Math. 149(2), 439 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Khater, A.H., Temsah, R.S., Hassan, M.M.: A Chebyshev spectral collocation method for solving Burgers-type equations. J. Comput. Appl. Math. 222(2), 333 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deghan, M., Asgar, H., Mohammad, S.: The solution of coupled Burgers’ equations using Adomian-Pade technique. Appl. Math. Comput. 189, 1034 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Rashid, A., Ismail, A.I.B.: A fourier Pseudospectral method for solving coupled viscous Burgers’ equations. Comput. Methods Appl. Math. 9(4), 412 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mittal, R.C., Arora, G.: Numerical solution of the coupled viscous Burgers’ equation. Commun. Nonlinear Sci. Numer. Simulat. 16, 1304 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mokhtari, R., Toodar, A.S., Chegini, N.G.: Application of the generalized differential quadrature method in solving Burgers’ equations. Commun. Theor. Phys. 56(6), 1009 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Srivastava, V.K., Awasthi, M.K., Tamsir, M.: A fully implicit finite-difference solution to one dimensional coupled nonlinear Burgers’ equation. Int. J. Math. Comput. Sci. Eng. 7(4), 283 (2013)zbMATHGoogle Scholar
  13. 13.
    Srivastava, V.K., Awasthi, M.K., Tamsir, M., Singh, S.: An implicit finite-difference solution to one dimensional coupled Burgers’ equation. Asian-Eur. J. Math. 6(4), 1350058 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrodinger equations. J. Comput. Phys. 205, 72–97 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yan, J., Shu, C.-W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17, 27–47 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhao, G.Z., Yu, X.J., Wu, D.: Numerical solution of the Burgers’ equation by local discontinuous Galerkin method. Appl. Math. Comput. 216, 3671–3679 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cheichan, M.S., Kashkool, H.A., Gao, F.: A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in one dimension. Int. J. Appl. Comput. Math. 5, 1–15 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhang, T., Tang, L.X.: A weak finite element method for elliptic problems in one space dimension. Appl. Math. Comput. 280, 1–10 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chen, Y., Zhang, T.: A weak Galerkin finite element method for Burgers’ equation. J. Comput. Appl. Math. 384, 103–119 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nee, J., Duan, J.: Limit set of trajectories of the coupled viscous Burgers’ equations. Appl. Math. Lett. 11(1), 57 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Korean Society for Informatics and Computational Applied Mathematics 2020

Authors and Affiliations

  1. 1.College of Education for Pure SciencesUniversity of BasrahBasrahIraq

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