A Finite Difference Approximation for Numerical Simulation of 2D Viscous Coupled Burgers Equations

Yıl 2022, Cilt: 10 Sayı: 3, 146 - 158, 09.09.2022

Murat Yağmurlu Abdulnasır Gagir

https://doi.org/10.36753/mathenot.981131

Cited By: 2

Öz

Many of the physical phenomena in nature are usually expressed in terms of algebraic, differential or integral equations.Several nonlinear phenomena playing a very important role in engineering sciences, physics and computational mathematics are usually modeled by those non-linear partial differential equations (PDEs). It is usually difficult and problematic to examine and find out nalytical solutions of initial-boundary value problems consisting of PDEs. In fact, there is no a certain method or technique working well for all these type equations. For this reason, their approximate solutions are usually preferred rather than analytical ones of such type equations. Thus, many researchers are concentrated on approximate methods and techniques to obtain numerical solutions of non-linear PDEs. In the present article, the numerical simulation of the two-dimensional coupled Burgers equation (2D-cBE) has been sought by finite difference method based on Crank-Nicolson type approximation. Widely used three test examples given with appropriate initial and boundary conditions are used for the simulation process. During the simulation process,the error norms $L_{2}$, $L_{\infty}$ are calculated if the exact solutions are already known, otherwise the pointwise values and graphics are provided for comparison. The newly obtained error norms $L_{2}$, $L_{\infty}$ by the presented schemes are compared with those of some of the numerical solutions in the literature. A good consistency and accuracy are observed both by numerical values and visual illustrations.

Anahtar Kelimeler

Two-dimensional viscous Burgers equation, Crank-Nicolson, Finite difference method

Kaynakça

• [1] Fletcher, C. A. J.: Generating exact solutions of the two-dimensional Burgers’ equations. International Journal for Numerical Methods in Fluids. 3, 213-216 (1983). https://doi.org/10.1002/fld.1650030302
• [2] Yagmurlu, N. M., Gagir A.: Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Mathematical Sciences and Applications E-notes. 9 (4), 158-169 (2021). https://doi.org/10.36753/mathenot.947552
• [3] 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). https://doi.org/10.1016/0021- 9991(83)90085-2
• [4] Goyon, O.: Multilevel Schemes for Solving Unsteady Equations. International Journal for Numerical Methods in Fluids. 22 (10), 937-959 (1996).
• [5] Arshed A., Siraj-ul-Islam, Sirajul H.: 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). https://doi.org/10.1080/15502280903108016
• [6] Jain, P. C., Holla, D. N.: Numerical solutions of coupled Burgers’ equation. International Journal of Non-Linear. Meechanics. 13 (4), 213-222 (1978), https://doi.org/10.1016/0020-7462(78)90024-0.
• [7] Bahadır, A. R.: A fully implicit finite-difference scheme for two-dimensional Burgers’ equations. Applied Mathematics and Computation. 137 (1), 131–137 (2003). https://doi.org/10.1016/S0096-3003(02)00091-7
• [8] Khater, A. H., Temsah, R. S, Hassan, M.M.: Chebyshev spectral collocation method for solving Burgers’-type equations. Journal of Computational and Applied Mathematics. 222 (2) 333–350 (2008). https://doi.org/10.1016/j.cam.2007.11.007https://doi.org/10.1016/j.cam.200711.007
• [9] Mittal, R. C., Jiwari, R.: Differential Quadrature Method for Two-Dimensional Burgers’ Equations. Interna- tional Journal for Computational Methods in Engineering Science and Mechanics. 10 (6), 450–459 (2009). https://doi:10.1080/15502280903111424
• [10] Liao, W.: A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations. Interna- tional Jlournal Numerical Methods in Fluids. 64 (5) 565–590 (2010). https://doi.org/10.1002/fld.2163
• [11] Zhu, H., Shu, H., Ding, M.: Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method. Computers and Mathematics with Applications. 60 (3) 840-848 (2010). https://doi.org/10.1016/j.camwa.2010.05.031
• [12] Srivastava, V. K., Tamsir, M., Bhardwaj, U., Sanyasiraju, Y.: Crank-Nicolson Scheme for Numerical Solutions of Two-dimensional Coupled Burgers’ Equations. International Journal of Scientific & Engineering Research. 2 (5), 1-6 (2011).
• [13] Tamsir, M., Srivastava, V. K.: A semi-implicit finite-difference approach for two-dimensional coupled Burgers equations. International Journal of Scientific & Engineering Research. 2 (6), 46-51 (2011). ISSN 2229-5518
• [14] Srivastava, V. K., Tamsir, M.: Crank-Nicolson Semi-Implicit Approach For Numerical Solutions of Two- Dimensional Coupled Nonlinear Burgers Equations. International Journal of Applied Mechanics and Engineering. 17 (2), 571-581 (2012).
• [15] Thakar, S., Wani, S.: Linear Method For Two Dimensional Burgers Equation. Ultra Scientist. 25 (1)A, 156-168 (2013).
• [16] Srivastava, V. K., Awasthi, M.K., Singh, S.: An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers’ equation. AIP Advances. 3 (12), 122105 (2013). https://doi:10.1063/1.4842595
• [17] Srivastava, V. K., Singh, S., Awasthi, M. K.: Numerical solutions of coupled Burgers equations by an implicit finite difference scheme. AIP Advances. 3 (8), 082131 (2013). https://doi: 10.1063/1.4820355
• [18] Srivastava, V. K., Singh, B. K.: A robust finite difference scheme for the numerical solutions of two dimensional time dependent coupled nonlinear Burgers equations. International Journal of Applied Mathematics and Mechanics. 10 (7), 28-39 (2014).
• [19] Zhang, L., Wang, L., Ding, X.:Exact finite-difference scheme and nonstandard finite-difference scheme for coupled Burgers equation. Advances in Difference Equations. 122 (2014) (2014). https://doi:10.1186/1687-1847-2014-122
• [20] Mittal, R. C., Tripathi, A.: Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements. Engineering Computations. 32 (5), 1275 - 1306.(2015). https://doi.org/10.1108/EC-04-2014-0067
• [21] Tamsir, M., Srivastava, V. K., Jiwari, R.: An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation. Applied Mathematics and Computation. 290, 111–124 (2016). https://doi.org/10.1016/j.amc.2016.05.048
• [22] Zhanlav, T., Chuluunbaatar, O., Ulziibayar, V.: Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers Equations. American Journal of Computational Mathematics. 6 (2), 120-129 (2016). https://doi:10.4236/ajcm.2016.62013
• [23] Ngondiep, E.: An efficient three-level explicit time-split scheme for solving two-dimensional unsteady nonlinear coupled Burgers’ equations. Int J Numer Meth Fluids. 92 (4), 266–284 (2020). https://doi.org/10.1002/fld.4783
• [24] Saqib, M., Hasnain, S., Mashat, D. S.: Highly Efficient Computational Methods for Two Dimen- sional Coupled Nonlinear Unsteady Convection-Diffusion Problems. IEEE Access. 5, 7139-7148 (2017). https://doi:10.1109/ACCESS.2017.2699320
• [25] Wubs, F. W., de Goede, E. D.: An explicit-implicit method for a class of time-dependent partial differential equations. Appl. Numer. Math. 9 (2), 157-181 (1992). https://doi.org/10.1016/0168-9274(92)90012-3
• [26] Kutluay,S.,Yag ̆murlu,N.M.:TheModifiedBi-quinticB-SplinesforSolvingtheTwo-DimensionalUnsteadyBurgers’ Equation. European International Journal of Science and Technology. 1(2), 23-39 (2012).
• [27] Bas ̧han,A.:AnumericaltreatmentofthecoupledviscousBurgers’equationinthepresenceofverylargeReynoldsnumber. Physica A: Statistical Mechanics and its Applications. 545, (2020). https://doi.org/10.1016/j.physa.2019.123755
• [28] Bas ̧han,A.,Karakoç,S.B.G.,Geyikli,T.:B-splineDifferentialQuadratureMethodfortheModifiedBurgers’Equation. Çankaya University Journal of Science and Engineering. 12 (1), 001–013 (2015) .
• [29] Uçar, Y., Yag ̆murlu, N. M., Bas ̧han, A.: Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-Spline Differential Quadrature Methods. Sigma J Eng & Nat Sci. 37 (1), 129-142 (2019).
• [30] Karakoç, S. B. G., Bas ̧han, A., Geyikli, T.: Two Different Methods for Numerical Solution of the Modified Burgers’ Equation. The Scientific World Journal. 2014 (5), 1-13 (2014). https://doi.org/10.1155/2014/780269.
• [31] Bas ̧han, A.: Nonlinear dynamics of the Burgers’ equation and numerical experiments. Math Sci (2021). https://doi.org/10.1007/s40096-021-00410-8.
• [32] Karakoç, B., Bhowmik, S. K.: Galerkin Finite Element Solution for Benjamin-Bona-Mahony-Burgers Equation with Cubic B-Splines. Computers and Mathematics with Applications. 77 (7), 1917–1932 (2019).
• [33] Bhowmik, S. K., Karakoç, S. B. G.: Numerical approximation of the generalized regularized long wave equation using Petrov Galerkin finite element method. Numerical Methods for Partial Differential Equations. 35 (6), 2236–2257 (2019).
• [34] Chai, Y., Ouyang, J.: Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equa- tions at high Reynolds numbers. Computers and Mathematics with Applications. 79 (5), 1287–1301 (2020). https://doi.org/10.1016/j.camwa.2019.08.036
• [35] Shukla, H. S., Tamsir, M., Srivastava, V. K., Kumar, J.: Numerical Solution of two dimensional coupled viscous Burgers’ Equation using the Modified Cubic B-Spline Differential Quadrature Method. AIP Advances. 4 (11), 1-10 (2014).https://doi.org/10.1063/1.4902507