Author Topic: Crank-Nicolson Scheme for Numerical Solutions of Two-dimensional Coupled Burgers  (Read 2298 times)

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Author : Vineet Kumar Srivastava, Mohammad Tamsir, Utkarsh Bhardwaj, YVSS Sanyasiraju
International Journal of Scientific & Engineering Research, Volume 2, Issue 5, May-2011
ISSN 2229-5518
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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.
Index Terms — Burgers’ equations; Crank-Nicolson scheme; finite- difference; Newton’s method; Reynolds number.

THE nonlinear coupled Burgers’ equation is a special form of incompressible Navier-Stokes equation without having pressure term and continuity equation. Burgers’ equation is an important partial differential equation from fluid dynamics, and is widely used for various physical applications, such as modeling of gas dy-namics and traffic flow, shock waves [1], investigating the shallow water waves[2,3], in examining the chemical reaction-diffusion model of Brusselator[4] etc. It is also used to test several numerical algorithms. The first attempt to solve Burgers’ equation analytically was given by Bateman [5], who derived the steady solution for a simple one-dimensional Burgers’ equation, which was used by J.M. Burger in [6] to model turbulence. In the past several years, numerical solution to one-dimensional Burgers’ equation and system of multidimensional Burgers’ equations have attracted a lot of attention and which has resulted in various finite-difference, finite-element and boundary element methods. Finite difference methods can be classified in two broad categories, i.e. explicit and implicit. Chabak and Sharma [7] have executed the solution of two dimensional coupled wave eqution explicitly. An implicit approach has been utilized for solving two dimensional coupled Burgers’ equations. Since in this paper the focus is numerical solutions of the two-dimensional Burgers’ equations, a detailed survey of the numerical schemes for solving the one-dimensional Burgers’ equation is not necessary. Interested readers can refer to [8-14] for more details.
Consider two-dimensional coupled nonlinear Burgers’ equations subject to the initial conditions and boundary conditions Where  is its boundary;  and  are the velocity components to be determined, , ,  and  are known functions and  is the Reynolds number.
The analytic solution of eqns. (1) and (2) was proposed by Fletcher using the Hopf-Cole transformation [15]. The numerical solutions of this system of equations have been solved by many researchers. Jain and Holla [16] developed two algorithms based on cubic spline method. Fletcher [17] has discussed the comparison of a number of different numerical approaches.Wubs and Goede [18] have applied an explicit–implicit method. Goyon [19] used several multilevel schemes with ADI.  Recently A. R. Bahadır [20] has applied a fully implicit method.
In this paper, Crank-Nicolson finite-difference scheme is used for solving two-dimensional coupled nonlinear Burgers’ equations. Two numerical examples have been carried out and their results are presented to illustrate the efficiency of the proposed method.
The computational domain  is discretized with uniform grid. Denote the discrete approximation of  and  at the grid point  by   and   respective-ly  where   is the grid size in x-direction,   is the grid size in y-direction, and  represents the increment in time.
Crank-Nicolson finite-difference approximation to (1) is given by Similarly, Crank-Nicolson approximation to (2) is given by Consider the above nonlinear system of equations in the form
Where, and   Newton’s method when it is ap-plied to (3) results in the following steps:
1.   set , an initial approximation.
2.   while for  until convergence do:
•   solve 
•   set 

Where  is the Jacobian  matrix and   is the correction vector. It is a   square matrix whose elements are blocks of size . Newton’s iteration at each time-step is stopped when .
By using Taylor’s series expansion one can easily see that the present scheme has accuracy of or-der .

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