A Fully Implicit Finite-Difference Solution to One Dimensional Coupled Nonlinear Burgers’ Equations
A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.
[1] S. E. Esipov, ”Coupled Burgers’ equations: a model of polydispersive
sedimentation”, Phys Rev E, Vol. 52 pp. 3711-3718, 1995.
[2] J. M. Burgers, ”A mathematical model illustrating the theory of turbulence”,
Adv. Appl. Mech., Vol. I, pp. 171-199, 1948.
[3] J. D. Cole, ”On a quasilinear parabolic equations occurring in aerodynamics”,
Quart. Appl. Math., Vol. 9, pp. 225-236, 1951.
[4] J. Nee and J. Duan, ”Limit set of trajectories of the coupled viscous
Burgers’ equations”, Appl. Math. Lett, Vol. 11, no. 1, pp. 57-61, 1998.
[5] D. Kaya, ”An explicit solution of coupled viscous Burgers’ equations
by the decomposition method”, J.J.M.M.S., Vol. 27, no. 11, pp. 675-680,
2001.
[6] A. A. Soliman, ”The modified extended tanh-function method for solving
Burgers-type equations”, Physica A, Vol. 361, pp. 394-404, 2006.
[7] M. A. Abdou and A. A. Soliman, ”Variational iteration method for solving
Burgers and coupled Burgers equations”, J. Comput. Appl. Math., Vol.
181, no. 2, pp. 245-251, 2005.
[8] G. W. Wei and Y. Gu, ”Conjugate filter approach for solving Burgers’
equation”, J. Comput. Appl. Math., Vol. 149, no. 2, pp. 439-456, 2002.
[9] A. H. Khater, R. S. Temsah, and M. M. Hassan, ”A Chebyshev spectral
collocation method for solving Burgers-type equations”, J. Comput. Appl.
Math., Vol. 222, no.2, pp. 333-350, 2008.
[10] M. Deghan, H. Asgar and S. Mohammad, ”The solution of coupled
Burgers’ equations using Adomian-Pade technique”, Appl. Math. Comput.,
Vol. 189, pp. 1034-1047, 2007.
[11] A. Rashid and A. I. B. Ismail, ”A fourierPseudospectral method for
solving coupled viscous Burgers’ equations”, Comput. Methods Appl.
Math., Vol. 9, no. 4, pp. 412-420, 2009.
[12] R. C. Mittal and G. Arora, ”Numerical solution of the coupled viscous
Burgers’ equation”, Commun.Nonlinear Sci. Numer.Simulat., Vol. 16, pp.
1304-1313, 2011.
[13] R. Mokhtari, A. S. Toodar and N. G. Chegini, ”Application of the
generalized differential quadrature method in solving Burgers’ equations”,
Commun. Theor. Phys., Vol. 56, no.6, pp. 1009-1015, 2011.
[14] S. Kutley, A. R. Bahadir and A. Ozdes, ”Numerical solution of onedimensional
Burgers’ equation: explicit and exact-explicit finite difference
methods”, J. Compt. Appl. Math, Vol. 103, pp. 251-261, 1999.
[15] M. K. Kadalbajoo and A. Awasthi, ”A numerical method based on
Crank-Nicolson scheme for Burgers’ equation”, Appl. Math. Compt., Vol.
182, pp. 1430-1442, 2006.
[16] P. C. Jain and D. N. Holla, ”Numerical solution of coupled Burgers’
equations”, Int. J. Numer. Meth. Eng., Vol. 12,pp. 213-222, 1978.
[17] C. A. J. Fletcher, ”A comparison of finite element and finite difference
of the one and two-dimensional Burgers’ equations”, J. Comput. Phys.,
Vol. 51, pp. 159-188, 1983.
[18] F. W. Wubs and E. D. de Goede, ”An explicit-implicit method for a class
of time-dependent partial differential equations”, Appl. Numer. Math., Vol.
9, pp. 157-181, 1992.
[19] O. Goyon, ”Multilevel schemes for solving unsteady equations”, Int. J.
Numer. Meth. Fluids, Vol. 22, pp. 937-959, 1996.
[20] A. R. Bahadir, ”A fully implicit finite-difference scheme for twodimensional
Burgers’ equation”, Applied Mathematics and Computation,
Vol. 137, pp. 131-137, 2003.
[21] V. K. Srivastava, M. Tamsir, U. Bhardwaj and Y. V. S. S. Sanyasiraju,
”Crank-Nicolson scheme for numerical solutions of two dimensional
coupled Burgers’ equations”, Int. J. Sci. Eng. Research, Vol. 2, no. 5,
pp. 1-6, 2011.
[22] M. Tamsir and V. K. Srivastava, ”A semi-implicit finite-difference
approach for two-dimensional coupled Burgers’ equations”, Int. J. Sci.
Eng. Research, Vol. 2, no. 6, pp. 1-6, 2011.
[23] V. K. Srivastava,Ashutosh and M. Tamsir, ”Generating exact solution
of three-dimensional coupled unsteady nonlinear generalized viscous
Burgers’ equations”, Int. J. Mod. Math. Sci., Vol. 5, no. 1, pp. 1-13,
2013.</p>
[1] S. E. Esipov, ”Coupled Burgers’ equations: a model of polydispersive
sedimentation”, Phys Rev E, Vol. 52 pp. 3711-3718, 1995.
[2] J. M. Burgers, ”A mathematical model illustrating the theory of turbulence”,
Adv. Appl. Mech., Vol. I, pp. 171-199, 1948.
[3] J. D. Cole, ”On a quasilinear parabolic equations occurring in aerodynamics”,
Quart. Appl. Math., Vol. 9, pp. 225-236, 1951.
[4] J. Nee and J. Duan, ”Limit set of trajectories of the coupled viscous
Burgers’ equations”, Appl. Math. Lett, Vol. 11, no. 1, pp. 57-61, 1998.
[5] D. Kaya, ”An explicit solution of coupled viscous Burgers’ equations
by the decomposition method”, J.J.M.M.S., Vol. 27, no. 11, pp. 675-680,
2001.
[6] A. A. Soliman, ”The modified extended tanh-function method for solving
Burgers-type equations”, Physica A, Vol. 361, pp. 394-404, 2006.
[7] M. A. Abdou and A. A. Soliman, ”Variational iteration method for solving
Burgers and coupled Burgers equations”, J. Comput. Appl. Math., Vol.
181, no. 2, pp. 245-251, 2005.
[8] G. W. Wei and Y. Gu, ”Conjugate filter approach for solving Burgers’
equation”, J. Comput. Appl. Math., Vol. 149, no. 2, pp. 439-456, 2002.
[9] A. H. Khater, R. S. Temsah, and M. M. Hassan, ”A Chebyshev spectral
collocation method for solving Burgers-type equations”, J. Comput. Appl.
Math., Vol. 222, no.2, pp. 333-350, 2008.
[10] M. Deghan, H. Asgar and S. Mohammad, ”The solution of coupled
Burgers’ equations using Adomian-Pade technique”, Appl. Math. Comput.,
Vol. 189, pp. 1034-1047, 2007.
[11] A. Rashid and A. I. B. Ismail, ”A fourierPseudospectral method for
solving coupled viscous Burgers’ equations”, Comput. Methods Appl.
Math., Vol. 9, no. 4, pp. 412-420, 2009.
[12] R. C. Mittal and G. Arora, ”Numerical solution of the coupled viscous
Burgers’ equation”, Commun.Nonlinear Sci. Numer.Simulat., Vol. 16, pp.
1304-1313, 2011.
[13] R. Mokhtari, A. S. Toodar and N. G. Chegini, ”Application of the
generalized differential quadrature method in solving Burgers’ equations”,
Commun. Theor. Phys., Vol. 56, no.6, pp. 1009-1015, 2011.
[14] S. Kutley, A. R. Bahadir and A. Ozdes, ”Numerical solution of onedimensional
Burgers’ equation: explicit and exact-explicit finite difference
methods”, J. Compt. Appl. Math, Vol. 103, pp. 251-261, 1999.
[15] M. K. Kadalbajoo and A. Awasthi, ”A numerical method based on
Crank-Nicolson scheme for Burgers’ equation”, Appl. Math. Compt., Vol.
182, pp. 1430-1442, 2006.
[16] P. C. Jain and D. N. Holla, ”Numerical solution of coupled Burgers’
equations”, Int. J. Numer. Meth. Eng., Vol. 12,pp. 213-222, 1978.
[17] C. A. J. Fletcher, ”A comparison of finite element and finite difference
of the one and two-dimensional Burgers’ equations”, J. Comput. Phys.,
Vol. 51, pp. 159-188, 1983.
[18] F. W. Wubs and E. D. de Goede, ”An explicit-implicit method for a class
of time-dependent partial differential equations”, Appl. Numer. Math., Vol.
9, pp. 157-181, 1992.
[19] O. Goyon, ”Multilevel schemes for solving unsteady equations”, Int. J.
Numer. Meth. Fluids, Vol. 22, pp. 937-959, 1996.
[20] A. R. Bahadir, ”A fully implicit finite-difference scheme for twodimensional
Burgers’ equation”, Applied Mathematics and Computation,
Vol. 137, pp. 131-137, 2003.
[21] V. K. Srivastava, M. Tamsir, U. Bhardwaj and Y. V. S. S. Sanyasiraju,
”Crank-Nicolson scheme for numerical solutions of two dimensional
coupled Burgers’ equations”, Int. J. Sci. Eng. Research, Vol. 2, no. 5,
pp. 1-6, 2011.
[22] M. Tamsir and V. K. Srivastava, ”A semi-implicit finite-difference
approach for two-dimensional coupled Burgers’ equations”, Int. J. Sci.
Eng. Research, Vol. 2, no. 6, pp. 1-6, 2011.
[23] V. K. Srivastava,Ashutosh and M. Tamsir, ”Generating exact solution
of three-dimensional coupled unsteady nonlinear generalized viscous
Burgers’ equations”, Int. J. Mod. Math. Sci., Vol. 5, no. 1, pp. 1-13,
2013.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63046", author = "Vineet K. Srivastava and Mukesh K. Awasthi and Mohammad Tamsir", title = "A Fully Implicit Finite-Difference Solution to One Dimensional Coupled Nonlinear Burgers’ Equations", abstract = "A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.
", keywords = "Burgers’ equation, Implicit Finite-difference method,
Newton’s method, Gauss elimination with partial pivoting.", volume = "7", number = "4", pages = "640-6", }
