High-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations. (English) Zbl 1540.65349
Summary: In this paper a high-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations is presented. The fifth-order weighted compact nonlinear scheme is used for the spatial discretization, while the third-order diagonal implicit Runge-Kutta method is used for the time discretization. The generated nonlinear system is solved by the Jacobian-free Newton-Krylov nonlinear solver, which is composed of the outer Newton iteration method and the inner Krylov subspace iteration method. Stability analysis shows that the presented implicit weighted compact nonlinear scheme is unconditionally stable. Numerical results indicate that the implicit scheme can achieve the designed third-order accuracy in time and has a great advantage in the computation efficiency compared to the third-order explicit total variation diminishing Runge-Kutta weighted essentially non-oscillatory scheme. In addition, the implicit scheme can capture discontinuities and shock waves with high resolution and can solve Burgers’ equations with all kinds of Reynolds numbers.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
high-order implicit method; weighted compact nonlinear scheme; Jacobian-free Newton-Krylov method; coupled viscous Burgers’ equations; shock-capturing capabilityReferences:
| [1] | Abbasbandy, S.; Tan, Y.; Liao, S. J., Newton-homotopy analysis method for nonlinear equations, Appl. Math. Comput., 188, 1794-1800 (2007) · Zbl 1119.65032 |
| [2] | Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math., 181, 245-251 (2005) · Zbl 1072.65127 |
| [3] | Aksan, E. N., Quadratic B-spline finite element method for numerical solution of the Burgers’ equation, Appl. Math. Comput., 174, 884-896 (2006) · Zbl 1090.65108 |
| [4] | Alexande, R., Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s, SIAM J. Numer. Anal., 14, 1006-1021 (1977) · Zbl 0374.65038 |
| [5] | Ali, A.; Siraj-ul Islam; Haq, S., A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers’ equations, Int. J. Comput. Methods Eng. Sci. Mech., 10, 406-422 (2009) · Zbl 1423.35303 |
| [6] | Bahadir, A. R.; Saglam, M., A mixed finite difference and boundary element approach to one-dimensional Burgers’ equation, Appl. Math. Comput., 160, 663-673 (2005) · Zbl 1062.65088 |
| [7] | Bahadr, A. R., A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Appl. Math. Comput., 137, 131-137 (2003) · Zbl 1027.65111 |
| [8] | Bak, S.; Kim, P.; Kim, D., A semi-Lagrangian approach for numerical simulation of coupled Burgers’ equations, Commun. Nonlinear Sci., 69, 31-44 (2019) · Zbl 1524.35327 |
| [9] | Botti, L., A choice of forcing terms in inexact Newton iterations with application to pseudo-transient continuation for incompressible fluid flow computations, Appl. Math. Comput., 266, 713-737 (2015) · Zbl 1410.76156 |
| [10] | Burgers, J. M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1, 171-199 (1948) |
| [11] | Chai, Y.; Ouyang, J., Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers, Comput. Math. Appl., 79, 1287-1301 (2020) · Zbl 1443.65198 |
| [12] | Chen, B. Y.; He, D. D.; Pan, K. J., A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers’ equation, Numer. Math. Theory Methods Appl., 11, 299-320 (2018) · Zbl 1424.65124 |
| [13] | Chen, C. K.; Zhang, X. H.; Liu, Z., A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system, Appl. Math. Comput., 372, Article 125009 pp. (2020) · Zbl 1433.65156 |
| [14] | Cleophas, K.; Benjamin, N.; John, W., Hybrid Crank-Nicolson-Du Fort and Frankel (CN-DF) scheme for the numerical solution of the 2-D coupled Burgers’ system, Appl. Math. Sci., 8, 2353-2361 (2014) |
| [15] | Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9, 225-236 (1951) · Zbl 0043.09902 |
| [16] | Deng, X. G.; Jiang, Y.; Mao, M. L.; Liu, H. Y.; L.i, S.; Tu, G. H., A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law, Comput. Fluids, 116, 29-45 (2015) · Zbl 1390.65065 |
| [17] | Deng, X. G.; Jiang, Y.; Mao, M. L.; Liu, H. Y.; Tu, G. H., Developing hybrid cell-edge and cell-node dissipative compact scheme for complex geometry flows, Sci. China Technol. Sci., 56, 2361-2369 (2013) |
| [18] | Deng, X. G.; Zhang, H. X., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165, 22-44 (2000) · Zbl 0988.76060 |
| [19] | Doan, K., Explicit solutions of generalized Boussinesq equations, J. Appl. Math., 1, 29-37 (2001) · Zbl 0976.35066 |
| [20] | Esipov, S. E., Coupled Burgers’ equations: a model of polydispersive sedimentation, Phys. Rev. E, 52, 3711-3718 (1995) |
| [21] | Fletcher, C. A.J., Generating exact solutions of the two-dimensional Burgers’ equations, Int. J. Numer. Methods Fluids, 3, 213-216 (1983) · Zbl 0563.76082 |
| [22] | Gao, Q.; Zou, M. Y., An analytical solution for two and three dimensional nonlinear Burgers’ equation, Appl. Math. Model., 45, 255-270 (2016) · Zbl 1446.35172 |
| [23] | Guo, Y.; Shi, Y. F.; Li, Y. M., A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation, Appl. Math. Comput., 281, 172-185 (2016) · Zbl 1410.65342 |
| [24] | Hopf, E., The partial differential equation ut+ uux= uxx, Commun. Pure. Appl. Math., 3, 201-230 (1950) · Zbl 0039.10403 |
| [25] | 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) · Zbl 1475.65125 |
| [26] | Jiang, Y. Q.; Chen, X.; Fan, R.; Zhang, X., High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations, Math. Comput. Simulation, 190, 607-621 (2021) · Zbl 1540.65298 |
| [27] | Kapoor, M.; Joshi, V., Numerical approximation of coupled 1D and 2D non-linear Burgers’ equations by employing modified quartic hyperbolic B-spline differential quadrature method, Int. J. Mech., 15, 37-55 (2021) |
| [28] | Kaya, D., An explicit solution of coupled viscous Burgers’ equation by the decomposition method, Int. J. Math. Math. Sci., 27, 675-680 (2001) · Zbl 0997.35077 |
| [29] | Khater, A. H.; Temsah, R. S.; Hassan, M. M., A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math., 222, 333-350 (2008) · Zbl 1153.65102 |
| [30] | Khoshfetrat, A.; Abedini, M. J., A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burgers’ equations in irregular domain, Appl. Math. Model., 36, 1885-1901 (2012) · Zbl 1243.65147 |
| [31] | Knoll, D. A.; Keyes, D. E., Jacobian-free Newton Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397 (2004) · Zbl 1036.65045 |
| [32] | Kumar, M.; Pandit, S., A composite numerical scheme for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Comm., 185, 809-817 (2014) · Zbl 1360.35117 |
| [33] | Kutluay, S.; Ucar, Y., Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method, Math. Methods Appl. Sci., 36, 2403-2415 (2013) · Zbl 1278.65153 |
| [34] | Liao, W. Y.; Zhu, J. P., Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation, Int. J. Comput. Math., 88, 2575-2590 (2011) · Zbl 1252.65141 |
| [35] | Mittal, R. C.; Arora, G., Numerical solution of the coupled viscous Burgers’ equation, Commun. Nonlinear Sci. Numer. Simul., 16, 1304-1313 (2011) · Zbl 1221.65264 |
| [36] | Mokhtari, R.; Toodar, A. S.; Chegini, N. G., Application of the generalized differential quadrature method in solving Burgers’ equations, Commun. Theor. Phys., 56, 1009-1015 (2011) · Zbl 1247.35108 |
| [37] | Mukundan, V.; Awasthi, A., A higher order numerical implicit method for non-linear Burgers’ equation, Differ. Equ. Dyn. Syst., 25, 169-186 (2017) · Zbl 1371.65098 |
| [38] | Rashid, A.; Ismail, A. I.B. M., A Fourier pseudospectral method for solving coupled viscous Burgers’ equations, Comput. Methods Appl. Math., 9, 412-420 (2009) · Zbl 1183.35245 |
| [39] | Rathan, S.; Raju, G. N., A modified fifth-order WENO scheme for hyperbolic conservation laws, Comput. Math. Appl., 75, 1531-1549 (2018) · Zbl 1409.65056 |
| [40] | Shao, L.; Feng, X. L.; He, Y. N., The local discontinuous Galerkin finite element method for Burgers’ equation, Math. Comput. Model., 54, 2943-2954 (2011) · Zbl 1235.65115 |
| [41] | Soliman, A. A., On the solution of two-dimensional coupled Burgers’ equations by variational iteration method, Chaos Solition Fract., 40, 1146-1155 (2009) · Zbl 1197.65203 |
| [42] | Srivastava, V. K.; Tamsir, M.; Awasthi, M. K.; Singh, S., One-dimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method, Aip. Adv., 4, 37119 (2014) |
| [43] | Tsai, C. C.; Shih, Y. T.; Lin, Y. T.; Wang, H. C., Tailored finite point method for solving one-dimensional Burgers’ equation, Int. J. Comput. Math., 94, 800-812 (2017) · Zbl 1364.65219 |
| [44] | Yan, Z. G.; Liu, H. Y.; Mao, M. L.; Zhu, H. J.; Deng, X. G., New nonlinear weights for improving accuracy and resolution of weighted compact nonlinear scheme, Comput. Fluids, 127, 226-240 (2016) · Zbl 1390.76639 |
| [45] | Zhang, X.; Jiang, Y. Q.; Chen, X.; Hu, Y. G., High-order implicit WCNS scheme for viscous Burgers equations, J. Numer. Methods Comput. Appl. (2021) |
| [46] | Zhang, S. H.; Jiang, S. F.; Shu, C. W., Development of nonlinear weighted compact schemes with increasingly higher order accuracy, J. Comput. Phys., 227, 7294-7321 (2008) · Zbl 1152.65094 |
| [47] | Zhao, G. Y.; Sun, M. B.; Xie, S. B.; Wang, H. B., Numerical dissipation control in an adaptive WCNS with a new smoothness indicator, Appl. Math. Comput., 330, 239-253 (2018) · Zbl 1427.76179 |
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