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.
Similar content being viewed by others
References
Zhang, R., Yu, X., Zhao, G.: Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations. Chin. Phys. B 20(11), 110205 (2011)
Kaya, D.: An explicit solution of coupled viscous Burgers’ equations by the decomposition method. JJMMS 27(11), 675 (2001)
Soliman, A.A.: The modified extended tanh-function method for solving Burgers-type equations. Physica A 361, 394 (2006)
Esipov, S.E.: Coupled Burgers’ equations: a model of polydispersive sedimentation. Phys. Rev. E 52, 3711 (1995)
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)
Wei, G.W., Gu, Y.: Conjugate filter approach for solving Burgers’ equation. J. Comput. Appl. Math. 149(2), 439 (2002)
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)
Deghan, M., Asgar, H., Mohammad, S.: The solution of coupled Burgers’ equations using Adomian-Pade technique. Appl. Math. Comput. 189, 1034 (2007)
Rashid, A., Ismail, A.I.B.: A fourier Pseudospectral method for solving coupled viscous Burgers’ equations. Comput. Methods Appl. Math. 9(4), 412 (2009)
Mittal, R.C., Arora, G.: Numerical solution of the coupled viscous Burgers’ equation. Commun. Nonlinear Sci. Numer. Simulat. 16, 1304 (2011)
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)
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)
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)
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)
Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)
Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrodinger equations. J. Comput. Phys. 205, 72–97 (2005)
Yan, J., Shu, C.-W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17, 27–47 (2002)
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)
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)
Zhang, T., Tang, L.X.: A weak finite element method for elliptic problems in one space dimension. Appl. Math. Comput. 280, 1–10 (2016)
Chen, Y., Zhang, T.: A weak Galerkin finite element method for Burgers’ equation. J. Comput. Appl. Math. 384, 103–119 (2016)
Nee, J., Duan, J.: Limit set of trajectories of the coupled viscous Burgers’ equations. Appl. Math. Lett. 11(1), 57 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hussein, A.J., Kashkool, H.A. Weak Galerkin finite element method for solving one-dimensional coupled Burgers’ equations. J. Appl. Math. Comput. 63, 265–293 (2020). https://doi.org/10.1007/s12190-020-01317-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-020-01317-8