Implicit extended discontinuous Galerkin scheme for solving singularly perturbed Burgers' equations
Abstract
We present the implicit-modal discontinuous Galerkin scheme for solving the coupled viscous and singularly perturbed Burgers’ equations. This scheme overcomes overshoot and undershoots phenomena in the singularly perturbed Burgers’ equations. We present the stability analysis and obtain suitable ranges for penalty terms and time steps. Also, we gain the constant of trace inequality for the approximate function and its first derivatives based on Legendre basis functions. The numerical results have good agreement with the analytical and available approximate solutions.
Keyword : discontinuous Galerkin method, backward Euler method, viscous Burgers’ equation, singularly perturbed Burgers’ equation, stability analysis
How to Cite
Khodayari-Samghabadi, S., Mondanizadeh, M., & Momeni-Masuleh, S. H. (2024). Implicit extended discontinuous Galerkin scheme for solving singularly perturbed Burgers’ equations. Mathematical Modelling and Analysis, 29(1), 1–22. https://doi.org/10.3846/mma.2024.16979
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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V.K. Srivastava, M. Tamsir, M.K. Awasthi and S. Singh. Onedimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method. AIP Adv., 4(3):037119, 2014. https://doi.org/10.1063/1.4869637
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M. Uzunca. Adaptive discontinuous Galerkin methods for non-linear reactive flows. Springer, 2016. https://doi.org/10.1007/978-3-319-30130-3
T. Warburton and J.S. Hesthaven. On the constants in hp-finite element trace inverse inequalities. Comput. Methods in Appl. Mech. Eng., 192(25):2765–2773, 2003. https://doi.org/10.1016/S0045-7825(03)00294-9
A.A. Alderremy, S. Saleem and F.A. Hendi. A comparative analysis for the solution of nonlinear Burgers’ equation. J Integr Neurosci., 14(3-4):503–523, 2018. https://doi.org/10.3233/JIN-180085
M. Baccouch and S. Kaddeche. Efficient Chebyshev pseudospectral methods for viscous Burgers’ equations in one and two space dimensions. Int. j. appl. math. comput., 5(1):18, 2019. https://doi.org/10.1007/s40819-019-0602-6
S. Bak, P. Kim and D. Kim. A semi-Lagrangian approach for numerical simulation of coupled Burgers’ equations. Commun Nonlinear Sci Numer Simul., 69:31–44, 2019. https://doi.org/10.1016/j.cnsns.2018.09.007
H.O. Bakodah, N.A. Al-Zaid, M. Mirzazadeh and Q. Zhou. Decomposition method for solving Burgers’ equation with Dirichlet and Neumann boundary conditions. Optik., 130:1339–1346, 2017. https://doi.org/10.1016/j.ijleo.2016.11.140
A. Bashan. A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number. Physica A: Statistical Mechanics and its Applications, 545:123755, 2020. https://doi.org/10.1016/j.physa.2019.123755
M. Bause and K. Schwegler. Higher order finite element approximation of systems of convection-diffusion-reaction equations with small diffusion. J. Comput. Appl. Math., 246:52–64, 2013. https://doi.org/10.1016/j.cam.2012.07.005
E.H. Doha, A.H. Bhrawy, M.A. Abdelkawy and R.M. Hafez. A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation. Eur. J. Phys., 12(2):111–122, 2014. https://doi.org/10.2478/s11534-014-0429-z
S. Gowrisankar and S. Natesan. An efficient robust numerical method for singularly perturbed Burgers’ equation. Appl. Math. Comput., 346:385–394, 2019. https://doi.org/10.1016/j.amc.2018.10.049
D. Kayao. An explicit solution of coupled viscous Burgers’ equation by the decomposition method. International Journal of Mathematics and Mathematical Sciences, 27(11):675–680, 2001. https://doi.org/10.1155/S0161171201010249
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(2):333–350, 2008. https://doi.org/10.1016/j.cam.2007.11.007
S. Khodayari-Samghabadi and S.H. Momeni-Masuleh. Implicit-modal discontinuous Galerkin scheme for two-phase flow with discontinuous capillary pressure. SIAM J. Sci. Comput., 40(4):B1131–B1160, 2018. https://doi.org/10.1137/17M1119937
M. Klinge and R. Weiner. Strong stability preserving explicit peer methods for discontinuous Galerkin discretizations. J. Sci. Comput., 75(2):1057–1078, 2018. https://doi.org/10.1007/s10915-017-0573-x
J.G.L. Laforgue and R.E. O’Malley Jr. Exponential asymptotics, the viscid Burgers’ equation, and standing wave solutions for a reaction-advection-diffusion model. Stud. Appl. Math., 102(2):137–172, 1999. https://doi.org/10.1111/14679590.00107
H. Lai and C. Ma. A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation. Physica A Stat. Mech., 395:445–457, 2014. https://doi.org/10.1016/j.physa.2013.10.030
Q. Li, Z. Chai and B. Shi. A novel lattice Boltzmann model for the coupled viscous Burgers’ equations. Appl. Math. Comput., 250:948–957, 2015. https://doi.org/10.1016/j.amc.2014.11.036
R.C. Mittal and G. Arora. Numerical solution of the coupled viscous Burgers’ equation. Commun. Nonlinear Sci. Numer. Simulat., 16(3):1304–1313, 2011. https://doi.org/10.1016/j.cnsns.2010.06.028
R.C. Mittal and R. Jiwari. A differential quadrature method for numerical solutions of Burgers’-type equations. Int. J. Numer. Methods Heat Fluid Flow, 22(7):880–895, 2012. https://doi.org/10.1108/09615531211255761
S. Park, P. Kim, Y.Jeon and S. Bak. An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework. Applied Mathematics and Computation, 428:127185, 2022. https://doi.org/10.1016/j.amc.2022.127185
C.S. Rao, P.L. Sachdev and M. Ramaswamy. Self-similar solutions of a generalized Burgers equation with nonlinear damping. Nonlinear Anal. Real World Appl., 4(5):723–741, 2003. https://doi.org/10.1016/S1468-1218(02)00083-4
A. Rashid and A.I.B. Ismail. A Fourier pseudospectral method for solving coupled viscous Burgers equations. Int. J. Comput. Methods, 9(4):412–420, 2009. https://doi.org/10.2478/cmam-2009-0026
B.D. Reddy. Introductory functional analysis: with applications to boundary value problems and finite elements, volume 27. Springer Science & Business Media, 1991.
V.K. Srivastava, M. Tamsir, M.K. Awasthi and S. Singh. Onedimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method. AIP Adv., 4(3):037119, 2014. https://doi.org/10.1063/1.4869637
B. Tripathi, A. Luca, S. Baskar, F. Coulouvrat and R. Marchiano. Element centered smooth artificial viscosity in discontinuous Galerkin method for propagation of acoustic shock waves on unstructured meshes. J. Comput. Phys., 366:298–319, 2018. https://doi.org/10.1016/j.jcp.2018.04.010
M. Uzunca. Adaptive discontinuous Galerkin methods for non-linear reactive flows. Springer, 2016. https://doi.org/10.1007/978-3-319-30130-3
T. Warburton and J.S. Hesthaven. On the constants in hp-finite element trace inverse inequalities. Comput. Methods in Appl. Mech. Eng., 192(25):2765–2773, 2003. https://doi.org/10.1016/S0045-7825(03)00294-9