A composite algorithm for numerical solutions of two-dimensional coupled Burgers’ equations. (English) Zbl 1477.65168
Summary: In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Software:
MatlabReferences:
| [1] | Fletcher, C. J., Computational Techniques for Fluid Dynamics (1991), Berlin, Germany: Springer, Berlin, Germany · Zbl 0717.76001 |
| [2] | Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quarterly of Applied Mathematics, 9, 3, 225-236 (1951) · Zbl 0043.09902 · doi:10.1090/qam/42889 |
| [3] | Burgers, J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 1, 171-199 (1948) · doi:10.1016/s0065-2156(08)70100-5 |
| [4] | Jain, P. C.; Holla, D. N., Numerical solutions of coupled Burgers’ equation, International Journal of Non-linear Mechanics, 13, 4, 213-222 (1978) · Zbl 0388.76049 · doi:10.1016/0020-7462(78)90024-0 |
| [5] | Fletcher, C. A. J., A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers’ equations, Journal of Computational Physics, 51, 1, 159-188 (1983) · Zbl 0525.65077 · doi:10.1016/0021-9991(83)90085-2 |
| [6] | Liao, W. Y., A fourth order finite-difference method for solving the system of two-dimensional Burgers’ equations, International Journal for Numerical Methods in Fluids, 64, 565-590 (2010) · Zbl 1377.65108 |
| [7] | Liu, F.; Weiping, S., Numerical solutions of two-dimensional Burgers’ equations by lattice Boltzmann method, Communications in Nonlinear Science and Numerical Simulation, 16, 1, 150-157 (2011) · Zbl 1221.76165 · doi:10.1016/j.cnsns.2010.02.025 |
| [8] | Zhao, G.; Yu, X.; Zhang, R., The new numerical method for solving the system of two-dimensional Burgers’ equations, Computers & Mathematics with Applications, 62, 8, 3279-3291 (2011) · Zbl 1232.65133 · doi:10.1016/j.camwa.2011.08.044 |
| [9] | Goyon, O., Multilevel schemes for solving unsteady equations, International Journal for Numerical Methods in Fluids, 22, 10, 937-959 (1996) · Zbl 0863.76043 · doi:10.1002/(sici)1097-0363(19960530)22:10<937::aid-fld387>3.0.co;2-4 |
| [10] | Wubs, F. W.; de Goede, E. D., An explicit-implicit method for a class of time-dependent partial differential equations, Applied Numerical Mathematics, 9, 2, 157-181 (1992) · Zbl 0749.65068 · doi:10.1016/0168-9274(92)90012-3 |
| [11] | Bahadir, A., A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Applied Mathematics and Computation, 137, 131-137 (2003) · Zbl 1027.65111 |
| [12] | EI-Sayed, M. S.; Kaya, D., On the numerical solution of the system of two-dimensional Burgers’ equations by decomposition method, Applied Mathematics and Computation, 158, 101-109 (2004) · Zbl 1061.65099 |
| [13] | Boules, A. N.; Eick, I. J., A spectral approximation of the two dimensional Burgers’ Equation, Indian Journal of Pure and Applied Mathematics, 34, 299-309 (2003) · Zbl 1031.35008 |
| [14] | Khater, A. H.; Temsah, R. S.; Hassan, M. M., A Chebyshev spectral collocation method for solving Burgers’-type equations, Journal of Computational and Applied Mathematics, 222, 245-251 (2008) · Zbl 1153.65102 · doi:10.1016/j.cam.2007.11.007 |
| [15] | Ali, A.; Siraj-ul-Islam, S.; Haq, S., A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10, 5, 406-422 (2009) · Zbl 1423.35303 · doi:10.1080/15502280903108016 |
| [16] | Siraj-ul-Islam, S.; Šarler, B.; Vertnik, R.; Kosec, G., Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations, Applied Mathematical Modelling, 36, 3, 1148-1160 (2012) · Zbl 1243.76076 · doi:10.1016/j.apm.2011.07.050 |
| [17] | Jiwari, R., A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Computer Physics Communications, 183, 11, 2413-2423 (2012) · Zbl 1302.35337 · doi:10.1016/j.cpc.2012.06.009 |
| [18] | Mittal, R. C.; Jiwari, R., A differential quadrature method for solving Burgers’-type equation, International Journal of Numerical Methods for Heat, 22, 1012, 880-895 (2016) · Zbl 1357.65220 |
| [19] | Mittal, R. C.; Jiwari, R., Differential quadrature method for two-dimensional Burgers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10, 6, 450-459 (2009) · Zbl 1423.35314 · doi:10.1080/15502280903111424 |
| [20] | Mittal, R. C.; Jiwari, R., Differential quadrature method for numerical solution of coupled viscous Burgers’ equations, International Journal of Computational Methods in Engineering Science Mechanics, 13, 1-5 (2012) · Zbl 07871313 · doi:10.1080/15502287.2011.654175 |
| [21] | Jiwari, R.; Mittal, R. C.; Sharma, K. K., A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation, Applied Mathematics and Computation, 219, 12, 6680-6691 (2013) · Zbl 1335.65070 · doi:10.1016/j.amc.2012.12.035 |
| [22] | Mittal, R. C.; Jiwari, R.; Sharma, K. K., A numerical scheme based on differential quadrature method to solve time dependent Burgers’ equation, Engineering Computations, 30, 117-131 (2013) |
| [23] | Shukla, H. S.; Tamsir, M.; Srivastava, V. K.; Kumar, J., Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method, AIP Advances, 4, 11 (2014) · doi:10.1063/1.4902507 |
| [24] | Lin, J.; Feng, W. J.; Reutskiy, S.; Ku, H. F.; He, Y. J., A semi-analytical method for solving a class of time fractional partial differential equations with variable coefficients, Applied Mathematics Letters, 112 (2021) · Zbl 1454.65126 |
| [25] | Lin, J.; Zhang, Y. H.; Reutskiy, S.; Feng, W. J., A novel meshless space-time backward substitution method and its application to nonhomogeneous advection-diffusion problems, Applied Mathematics and Computation, 398, 1 (2021) · Zbl 1508.65143 |
| [26] | Zhang, Y.; Lin, J.; Reutskiy, S.; Sun, H. G.; Fen, W. J., The improved backward substitution method for the simulation of time-dependent nonlinear coupled Burgers’ equations, Results in Physics, 18 (2020) |
| [27] | Shu, C., Differential Quadrature and its Application in Engineering (2000), Britain, UK: Springer-Verlag, Britain, UK · Zbl 0944.65107 |
| [28] | Bellman, R.; Kashef, B. G.; Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics, 10, 1, 40-52 (1972) · Zbl 0247.65061 · doi:10.1016/0021-9991(72)90089-7 |
| [29] | Korkmaz, A.; Dağ, İ., Shock wave simulations using sinc differential quadrature method, Engineering Computations, 28, 6, 654-674 (2011) · Zbl 1284.76292 · doi:10.1108/02644401111154619 |
| [30] | Korkmaz, A.; Dağ, İ., A differential quadrature algorithm for nonlinear Schrödinger equation, Nonlinear Dynamics, 56, 1-2, 69-83 (2009) · Zbl 1172.65381 · doi:10.1007/s11071-008-9380-0 |
| [31] | Jiwari, R.; Gupta, R. K.; Kumar, V., Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Ain Shams Engineering Journal, 5, 4, 1343-1350 (2014) · doi:10.1016/j.asej.2014.06.005 |
| [32] | Jiwari, R.; Pandit, S.; Mittal, R. C., Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Computer Physics Communications, 183, 3, 600-616 (2012) · Zbl 1264.65173 · doi:10.1016/j.cpc.2011.12.004 |
| [33] | Jiwari, R.; Pandit, S.; Mittal, R. C., A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions, Applied Mathematics and Computation, 218, 13, 7279-7294 (2012) · Zbl 1246.65174 · doi:10.1016/j.amc.2012.01.006 |
| [34] | Kumar, V.; Jiwari, R.; Kumar Gupta, R., Numerical simulation of two dimensional quasilinear hyperbolic equations by polynomial differential quadrature method, Engineering Computations, 30, 7, 892-909 (2013) · doi:10.1108/ec-02-2012-0030 |
| [35] | Lin, J.; Zhao, Y.; Watson, D.; Chen, C. S., The radial basis function differential quadrature method with ghost points, Mathematics and Computers in Simulation, 173, 105-114 (2020) · Zbl 1510.65310 · doi:10.1016/j.matcom.2020.01.006 |
| [36] | Abbas, M.; Majid, A. A.; Ismail, A. I. M.; Rashid, A., The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Applied Mathematics and Computation, 239, 74-88 (2014) · Zbl 1334.65163 · doi:10.1016/j.amc.2014.04.031 |
| [37] | Jiwari, R.; Singh, S.; Kumar, A., Numerical simulation to capture the pattern formation of coupled reaction-diffusion models, Chaos, Solitons & Fractals, 103, 422-439 (2017) · Zbl 1380.65317 · doi:10.1016/j.chaos.2017.06.023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
