Numerical Solution of two dimensional coupled viscous Burgers Equation using the Modified Cubic B Spline Differential Quadrature Method

In this paper, a numerical solution of the two dimensional nonlinear coupled viscous Burgers equation is discussed with the appropriate initial and boundary conditions using the modified cubic B spline differential quadrature method. In this method, the weighting coefficients are computed using the modified cubic B spline as a basis function in the differential quadrature method. Thus, the coupled Burgers equations are reduced into a system of ordinary differential equations (ODEs). An optimal five stage and fourth order strong stability preserving Runge Kutta scheme is applied to solve the resulting system of ODEs. The accuracy of the scheme is illustrated via two numerical examples. Computed results are compared with the exact solutions and other results available in the literature. Numerical results show that the MCB DQM is efficient and reliable scheme for solving the two dimensional coupled Burgers equation.


Introduction
Consider the two dimensional nonlinear unsteady coupled viscous Burgers' equations: 22 22 1 , Re

Modified cubic B-spline differential quadrature method
In 1972, Bellman et al. 16 introduced differential quadrature method (DQM). This method approximates the spatial derivatives of a function using the weighted sum of the functional values at the certain discrete points. In DQM, the weighting coefficients are determined using several kinds of test functions such as spline function 23 , sinc function 24 In the same way, the first and second order spatial partial derivatives of   ,, v x y t with respect to x and with respect to y are approximated as: are the weighting coefficients of the rth-order spatial partial derivatives with respect to x and y .
The cubic B-spline basis functions 22 at the knots are defined as: The values of cubic B-splines and its derivatives at the nodal points are depicted in Table 1.
Now, to get a diagonally dominant system of the linear equations, the cubic B-spline basis functions are modified as 25 : , we get a system of linear equations: With the help of Eq. (2.9), (2.10) and Table 1, Eq. (2.11) reduces into a tridiagonal system of equations: Here, we point out that the coefficient matrix A is invertible.

Results and discussion
Here we consider two test problems of two dimensional coupled Burgers equation as given in the introduction part to provide the MCB-DQM numerical solutions. The accuracy and consistency of the scheme is measured in terms of error norms 2 L and L  , defined as: and Dirichet boundary conditions:  Tables 2 and 3 along with the results given by Srivastava et al. [14][15] and Bahadir 9 at some typical grid point. The tabulated results show that the proposed scheme produces better result than Bahadir 9 . Tables   4 and 5 show the errors 2 L and L  , and also the rate of convergence of u and v components, respectively, at Re  100, t  1.0 for t  0.0001. From Tables 4 and 5, it can be observed that the MCB-DQM performs better than Srivastava et al. 15 and gives more than quadratic rate of convergence. The MCB-DQM computed solutions of u and v for Re  100 at t  0.5 are depicted in Fig. 1 while Fig. 2 shows analytical solutions of u and v , respectively.
For the test problem 4.2, numerical computations are carried out with the parameters: Re  50, 100, and 20 20  grid, t  0.625 for t  0.0001 in order to compare the computed results with those given by Jain & Holla 5 , Bahadir 9 , and Srivastava et al. [14][15] . Table 6 shows the comparisons of numerical results obtained using the MCB-DQM scheme at t  0.625, with the methods of Jain & Holla 5 , Bahadir 9 , and Srivastava et al. [14][15] . From Table 6, it can be noticed that the computed MCB-DQM results are in good agreement with Jain & Holla 5 , Bahadir 9 , and Srivastava et al. [14][15] . Fig. 3 depicts the MCB-DQM computed u and v solutions corresponding to Re =50, 100 and 500 at t  0.625.

Conclusions
A modified cubic B-spline differential quadrature method is presented for the numerical solutions of two dimensional nonlinear coupled viscous Burgers' equations. The computed results show that the solution obtained by this scheme is highly accurate and very close to the exact solutions. We also notice that the scheme has more than quadratic rate of convergance.
The obtained results show that the MCB-DQM is a promising numerical scheme for solving the higher dimensional nonlinear physical problems governed by partial differential equations.