Application‎ ‎of the‎ ‎Hybridized Discontinuous Galerkin Method for Solving One-Dimensional Coupled Burgers Equations

Document Type : Original Scientific Paper

Authors

1 ‎Department of Mathematical Sciences‎, ‎ ‎Isfahan University of Technology, ‎Isfahan‎, ‎I‎. ‎R‎. ‎Iran

2 ‎Department of Mathematics, ‎Tafresh University,‎Tafresh‎, ‎I‎. ‎R‎. ‎Iran

Abstract

‎This paper is devoted to proposing hybridized discontinuous Galerkin (HDG) approximations for solving a system of coupled Burgers equations (CBE) in a closed interval‎. ‎The noncomplete discretized HDG method is designed for a nonlinear weak form of one-dimensional $x-$variable such that numerical fluxes are defined properly‎, ‎stabilization parameters are applied‎, ‎and broken Sobolev approximation spaces are exploited in this scheme‎. ‎Having necessary conditions on the stabilization parameters‎, ‎it is proven in a theorem and corollary that the proposed method is stable with imposed homogeneous Dirichlet and/or periodic boundary conditions to CBE‎. ‎The desired HDG method is stated by using the Crank-Nicolson method for time-variable discretization and the Newton-Raphson method for solving nonlinear systems‎. ‎Numerical experiences show that the optimal rate of convergence is gained for approximate solutions and their first derivatives‎.

Keywords

Main Subjects

References

[1] S. E. Esipov, Coupled Burgers equations: a model of polydispersive sedimentation, Phys. Rev. E 52 (1995) 3711 -3718,
https://doi.org/10.1103/PhysRevE.52.3711.
[2] J. Nee and J. Duan, Limit set of trajectories of the coupled viscous Burgers’ equations, Appl. Math. Lett. 11 (1998) 57 - 61, https://doi.org/10.1016/S0893-9659(97)00133-X.
[3] R. Mokhtari, A. S. Toodar and N. G. Chegini, Application of the generalized differential quadrature method in solving Burgers’ equations, Commun. Theor. Phys. 56 (2011) #1009, https://doi.org/10.1088/0253-6102/56/6/06.
[4] R. Mokhtari and M. Mohseni, A meshless method for solving mKdV equation, Comput. Phys. Commun. 183 (2012) 1259 - 1268, https://doi.org/10.1016/j.cpc.2012.02.006.
[5] R. Mokhtari and S. Torabi Ziaratgahi, Numerical solution of RLW equation using integrated radial basis functions, Appl. Comput. Math. 10 (2011) 428-448.
[6] G. W.Wei and Y. Gu, Conjugate filter approach for solving Burgers’ equation, J. Comput. Appl. Math. 149 (2002) 439-456, https://doi.org/10.1016/S0377-0427(02)00488-0.
[7] A. H. Khater, R. S. Temsah and M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math. 222 (2008) 333 -350, https://doi.org/10.1016/j.cam.2007.11.007.
[8] M. Dehghan, A. Hamidi and M. Shakourifar, The solution of coupled Burgers’ equations using Adomian–Pade technique, Appl. Math. Comput. 189 (2007) 1034 - 1047, https://doi.org/10.1016/j.amc.2006.11.179.
[9] R. C. Mittal and G. Arora, Numerical solution of the coupled viscous Burgers’ equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1304 - 1313, https://doi.org/10.1016/j.cnsns.2010.06.028.
[10] V. K. Srivastava, M. K. Awasthi and M. Tamsir, A fully implicit finitedifference solution to one dimensional coupled nonlinear Burgers’ equations, Int. J. Math. Sci. 7 (2013) #23.
[11] V. K. Srivastava, M. K. Awasthi, M. Tamsir and S. Singh, An implicit finite difference solution to one dimensional coupled Burgers’ equation, Asian-Eur. J. Math. 6 (2013) #1350058.
[12] A. J. Hussein and H. A. Kashkool, Weak Galerkin finite element method for solving one-dimensional coupled Burgers’ equations, J. Appl. Math. Comput. 63 (2020) 265 - 293, https://doi.org/10.1007/s12190-020-01317-8.
[13] M. S. Fabien, A high-order implicit hybridizable discontinuous Galerkin method for the Benjamin-Bona-Mahony equation, J. Numer. Methods Fluids 93 (2021) 543 - 558, https://doi.org/10.1002/fld.4896.
[14] Z. Rong-Pei, Y. Xi-Jun and Z. Guo-Zhong, Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations, Chinese Phys. B 20 (2011) #110205, https://doi.org/10.1088/1674-1056/20/11/110205.
[15] N. Chegini and R. Stevenson, Adaptive wavelet schemes for parabolic problems: sparse matrices and numerical results, SIAM J. Numer. Anal. 49 (2011) 182 - 212, https://doi.org/10.1137/100800555.
[16] N. Chegini and R. Stevenson, An adaptive wavelet method for semi-linear first-order system least squares, Comput. Methods Appl. Math. 15 (2015) 439 - 463, https://doi.org/10.1515/cmam-2015-0023.
[17] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation (No. LA-UR-73-479; CONF-730414-2), Los Alamos Scientific Lab. N. Mex.(USA) (1973).
[18] B. Cockburn, S. Hou and C. W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990) 545 - 581, https://doi.org/10.2307/2008501.
[19] B. Cockburn and C. W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys. 141 (1998) 199 - 224, https://doi.org/10.1006/jcph.1998.5892.
[20] B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998) 2440 - 2463, https://doi.org/10.1137/S0036142997316712.
[21] B. Cockburn and J. Gopalakrishnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004) 283 - 301.
[22] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009) 1319 - 1365,
https://doi.org/10.1137/070706616.
[23] B. Cockburn, J. Gopalakrishnan and F. J. Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010) 1351 - 1367.
[24] B. Cockburn, J. Guzmán, S. C. Soon and H. K. Stolarski, An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM J. Numer. Anal. 47 (2009) 2686 - 2707, https://doi.org/10.1137/080726914.
[25] J. Kraus, P. L. Lederer, M. Lymbery, K. Osthues and J. Schöberl, Hybridized discontinuous Galerkin/hybrid mixed methods for a multiple network poroelasticity model with application in biomechanics, SIAM J. Sci. Comput. 45
(2023) B802 - B827, https://doi.org/10.1137/22M149764X.
[26] A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications, IMA J. Numer. Anal. 29 (2009) 827 - 855, https://doi.org/10.1093/imanum/drn038.
[27] S. Baharlouei, R. Mokhtari and N. Chegini, Solving two-dimensional coupled Burgers equations via a stable hybridized discontinuous Galerkin method, Iran. j. numer. anal. optim. 13 (2023) 397 - 425, https://doi.org/10.22067/IJNAO.2023.80916.1215.
Mathematics Interdisciplinary Research
Volume 9, Issue 4
December 2024
Pages 349-372

Files

Share

How to cite

Statistics