IJSER Home >> Journal >> IJSER
International Journal of Scientific and Engineering Research
ISSN Online 2229-5518
ISSN Print: 2229-5518 5    
Website: http://www.ijser.org
scirp IJSER >> Volume 2, Issue 5, May 2011 Edition
Crank-Nicolson scheme for numerical solutions of two-dimensional coupled Burgers’ equations
Full Text(PDF, 3000)  PP.  
Author(s)
Vineet Kumar Srivastava, Mohammad Tamsir, Utkarsh Bhardwaj, YVSS Sanyasiraju
KEYWORDS
Burgers’ equations; Crank-Nicolson scheme; finite- difference; Newton’s method; Reynolds number
ABSTRACT
The two-dimensional Burgers’ equation is a mathematical model to describe various kinds of phenomena such as turbulence and viscous fluid. In this paper, Crank-Nicolson finite-difference method is used to handle such problem. The proposed scheme forms a system of nonlinear algebraic difference equations to be solved at each time step. To linearize the non-linear system of equations, Newton’s method is used. The obtained linear system is then solved by Gauss elimination with partial pivoting. The proposed scheme is unconditionally stable and second order accurate in both space and time. Numerical results are compared with those of exact solutions and other available results for different values of Reynolds number. The proposed method can be easily implemented for solving nonlinear problems evolving in several branches of engineering and science.
References
[1] Cole JD, “On a quasilinear parabolic equations occurring in aerodynamics,” Quart Appl Math 1951; 9:225-36.

[2] J.D. Logan, ”An introduction to nonlinear partial differential equations, “ Wily-Interscience, New York, 1994.

[3] L. Debtnath, ”Nonlinear partial differential equations for scientist and engineers,” Birkhauser, Boston, 1997.

[4] G. Adomian, ”The diffusion-Brusselator equation,” Comput. Math. Appl. 29(1995) 1-3.

[5] Bateman H., “some recent researches on the motion of fluids,” Monthly Weather Review 1915; 43:163-170

[6] Burger JM, “A Mathematical Model Illustrating the Theory of Turbulence, ”Advances in Applied mathematics 1950; 3:201-230

[7] S.K. Chabak, P.K. Sharma, “Numerical simulation of coupled wave equation,” Int Review of Pure and Applied Mathematics 2007; 2(1):59-69.

[8] Hon YC, Mao XZ,”An efficient numerical scheme for Burgerslike equations,”Applied Mathematics and Computation 1998; 95:37-50.

[9] Aksan EN, Ozdes A, ”A numerical solution of Burgers’ equation,” Applied Mathematics and Computation 2004; 156:395-402.

[10] Mickens R, ”Exact solutions to difference equation models of Burgers’ equation,”Numerical Methods for Partial Differential Equations 1986; 2(2):123-129.

[11] Kutluay S, Bahadir AR, Ozdes A,”Numerical solution of onedimensional Burgers’ equation: explicit and exact-explicit finite difference methods,” Journal of Computational and Applied Mathematics 1999; 103:251-261.

[12] Kutluay S, Rsen A,”A linearized numerical scheme for Burgerslike equations,” Applied Mathematics and Computation 2004; 156:295-305.

[13] Ozis T, Aslan Y,” The semi-approximate approach for solving Burgers’ equation with high Reynolds number,”Applied Mathematics and Computation 2005; 163:131-145.

[14] Wenyuan Liao,” An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation,” Applied Mathematics and Computation 2008; 206:755-764.

[15] C.A.J. Fletcher, “Generating exact solutions of the twodimensional Burgers’ equation”, Int. J. Numer. Meth. Fluids 3 (1983) 213–216.

[16] P.C. Jain, D.N. Holla, “Numerical solution of coupled Burgers_ equations, “Int. J. Numer. Meth.Eng. 12 (1978) 213–222.

[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, (1983), 159-188.

[18] F.W. Wubs, E.D. de Goede, “An explicit–implicit method for a class of time-dependent partial differential equations, “Appl. Numer. Math. 9 (1992) 157–181.

[19] O. Goyon,” Multilevel schemes for solving unsteady equations,” Int. J. Numer. Meth. Fluids 22(1996) 937–959.

[20] Bahadir AR. “A fully implicit finite-difference scheme for twodimensional Burgers’ equation,” Appl Math Comput 2003; 137:131-7.

Untitled Page