A semiimplicit finitedifference approach for twodimensional coupled Burgers' equations

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Author(s) 
Mohammad Tamsir, Vineet Kumar Srivastava 

KEYWORDS 
Burgers’ equations; finite difference; semiimplicit scheme; Reynolds number.


ABSTRACT 
The twodimensional Burgers' equation is a mathematical model which is used to describe various kinds of phenomena such as turbulence and viscous fluid. In this paper, a semiimplicit finitedifference method is used to handle such problem. The proposed scheme forms a system of linear algebraic difference equations to be solved at each timestep. The linear system is solved by direct method. Numerical results are compared with those of exact solutions and other available results. The present method performs well. The proposed scheme can be extended for solving nonlinear problems arising in mechanics and other areas of engineering and science.


References 

[1] Cole JD, “On a quasilinear parabolic equations occurring in
aerodynamics,” Quart Appl Math 1951; 9:22536.
[2] J.D. Logan, ”An introduction to nonlinear partial differential
equations, “ WilyInterscience, New York, 1994.
[3] L. Debtnath, ”Nonlinear partial differential equations for scientist
and engineers,” Birkhauser, Boston, 1997.
[4] G. Adomian, ”The diffusionBrusselator equation,” Comput.
Math. Appl. 29(1995) 13.
[5] Bateman H., “some recent researches on the motion of fluids,”
Monthly Weather Review 1915; 43:163170
[6] Burger JM, “A Mathematical Model Illustrating the Theory of
Turbulence, ”Advances in Applied mathematics 1950; 3:201230
[7] T.Ozis, Y.Aslan, “The semiapproximate approach for solving
Burgers’ equation with high Reynolds number,” Appl. Math.
Comput. 163 (2005) 131145.
[8] Hon YC, Mao XZ,”An efficient numerical scheme for Burgerslike
equations,”Applied Mathematics and Computation 1998;
95:3750.
[9] Aksan EN, Ozdes A, ”A numerical solution of Burgers’ equation,”
Applied Mathematics and Computation 2004; 156:395402.
[10] Mickens R, ”Exact solutions to difference equation models of
Burgers’ equation,”Numerical Methods for Partial Differential
Equations 1986; 2(2):123129.
[11] Kutluay S, Bahadir AR, Ozdes A,”Numerical solution of onedimensional
Burgers’ equation: explicit and exactexplicit finite
difference methods,” Journal of Computational and Applied
Mathematics 1999; 103:251261.
[12] Kutluay S, Rsen A,”A linearized numerical scheme for Burgerslike
equations,” Applied Mathematics and Computation 2004;
156:295305.
[13] Wenyuan Liao,” An implicit fourthorder compact finite difference
scheme for onedimensional Burgers’ equation,” Applied
Mathematics and Computation 2008; 206:755764.
[14] C.A.J. Fletcher, “Generating exact solutions of the twodimensional
Burgers’ equation”, Int. J. Numer. Meth. Fluids 3
(1983) 213–216.
[15] P.C. Jain, D.N. Holla, “Numerical solution of coupled Burgers_
equations, “Int. J. Numer. Meth.Eng. 12 (1978) 213–222.
[16] C.A.J. Fletcher,” A comparison of finite element and finite difference
of the one and twodimensional Burgers’ equations,” J.
Comput. Phys., Vol. 51, (1983), 159188.
[17] F.W. Wubs, E.D. de Goede, “An explicit–implicit method for a
class of timedependent partial differential equations, “Appl.
Numer. Math. 9 (1992) 157–181.
[18] O. Goyon,” Multilevel schemes for solving unsteady equations,”
Int. J. Numer. Meth. Fluids 22(1996) 937–959.
[19] Bahadir AR. “A fully implicit finitedifference scheme for twodimensional
Burgers’ equation,” Appl Math Comput 2003;
137:1317.
[20] Vineet Kumar Srivastava, Mohammad Tamsir, Utkarsh Bhardwaj,
YVSS Sanyasiraju, “CrankNicolson scheme for numerical
solutions of two dimensional coupled Burgers’ equations,” International
Journal of Scientific & Engineering Research Volume
2, Issue 5, May2011.
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