A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. 'e developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. 'e established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. 'e technique can be extended for various multidimensional Burgers’ equations after some modifications.

Due to its extensive scope of applicability, various numerical schemes have been constructed to study its numerical solutions. Moreover, due to its application in various fields of science and technology, researchers and scientists are still interested in developing algorithms to find their numerical and exact solutions. A great number of works has been studied for finding approximate solutions of Burgers' equation, for example, cubic spline method [4], finite element and difference methods [5][6][7][8], multilevel alternating direction implicit schemes [9], and various explicit and implicit methods [10,11]. Furthermore, the decomposition method [12], spectral method [13], Chebyshev collocation method [14], and local discontinuous radial basis function collocation method [15,16] are investigated in literature. Also, Haar wavelet quasilinearization approach [17] and differential quadrature methods (DQMs) [18][19][20][21][22][23] have been developed. In recent years, new meshless methods [24][25][26] for various types of Burgers' equations have been developed.
In Lagrange interpolation-based DQMs [18][19][20][21][22], Lagrange's fundamentals are used to compute the weighting coefficients. In these cases, as the number of grid points increases the weights become unstable. Herein, to reduce this instability, the modified cubic trigonometric B-spline functions are used to the weighting coefficients of DQMs.
In this article, a new numerical algorithm is developed based on the finite difference and the modified cubic trigonometric B-spline (CTBS) DQMs for approximate solutions of coupled two-dimensional Burgers' equations' weighting coefficients (WCs) of DQM are calculated by using the modified CTBS functions as test functions which are different from the conventional technique of Lagrange interpolation [27]. Some well-known test problems are worked out to inspect the correctness and competence of the planned approach. e techniques lead to correct results with insignificant L ∞ , RMS and L 2 errors.

Differential Quadrature Method
Recently, DQMs have become popular for solving nonlinear partial differential equations (PDEs) arising in nonlinear phenomena. DQMs discretize the first and second derivatives over 1D domain Ω � [α, β] as follows: where α (1) ij and α (2) ij are unknown coefficients weighting the first and second derivatives, respectively, and x i , i � 1, 2, . . . , N, are uniform grids as well as nonuniform grids that exist in the domain. Bellman et al. [28] introduced two approaches to calculate WCs. Furthermore, to modify Bellman's approaches for finding WCs, many efforts have been carried out such as Lagrange interpolated cosine functions, spline functions, Legendre polynomials, Lagrange interpolation polynomials, and radial basis functions (see [19,[29][30][31][32][33][34][35] and the references therein) to determine these coefficients. In this study, we determine WCs with the use of CTBS functions after some modifications.

Cubic Trigonometric B-Spline Functions. In this section, we mesh the solution domain
Now, the piecewise CTBS basis functions τB j (x) over the uniform mesh are defined as follows [36,37]: e basis over the region α ≤ x ≤ β is formed by the set Every CTBS covers four elements. Now, with the help of Table 1, we have tabulated the values of B j (x) and its derivatives as follows: ,

Modified Cubic Trigonometric B-Spline Functions.
In this work, we compute WCs of DQM with the help of modified CTBS function defined in (6) as follows: It is worth mentioning that the modified functions τB j (x) , j � 0, 1, . . . , N, are linearly independent. On the solution domain [α, β], these functions create a family of basis functions.

Weighting Coefficients for Modified Cubic Trigonometric
B-Spline Differential Quadrature Method. Now, substitute the modified functions τB j (x) , for j � 0, 1, . . . , N, into equation (4). e matrix form of the equation is as follows: where A is (N + 1) × (N + 1) coefficient matrix: N, and B k at x k , for k � 0, 1, . . . , N, are as follows: Furthermore, with the help of omas algorithm WCs, α (1) ij are achieved as solutions of tridiagonal systems of equation (11). Similarly, with the help of the above method, it is easy to calculate second-order WCs β (2) ij .

Two-Dimensional Modified Cubic Trigonometric B-Spline
Differential Quadrature Method. In order to apply this method to 2D nonlinear problems, first of all, decompose the domain Ω � (x, y): and Δy � y j − y j−1 in x and y direction, respectively. is modified technique helps to estimate the 1 st order partial derivatives of u(x, y, t) at a point as follows: where α (1) ij is WCs for the 1 st order derivatives w.r.t. x. Similarly, β (1) jk are coefficients w.r.t. y. In order to compute the 2D WCs, we can define the functions τB j (y) , j � 0, 1, . . . , N, as in equation (10). Furthermore, take the test functions as T ij (x, y) � τB i (x)τB j (y). Now, with the help of the axioms of vector space and substituting the value of T ij (x, y) into equations (14) and (15), we have Furthermore, applying the well-known algorithm " omas algorithm" and proceeding with the same methods as in the case of equation (11), the solutions of the systems give the value of α (1) ij and β (1) jk . In 2D case, the WCs in higherorder derivatives can be considered as follows: where α (r) ij and β (r) ij are WCs for r th order partial derivatives w.r.t. x and y, respectively.

Numerical Algorithm for Two-Dimensional Coupled Burgers' Equation
In this section, the numerical algorithm is developed in the following sections.

Semidiscretization in Time.
Applying forward difference on time derivatives and weighted average on spatial derivatives, we have y, t + nΔt), Δt step length in time direction, and 0 ≤ θ ≤ 1. e nonlinear term is linearized in the following manner: with ICs and prescribed BCs (3). After simplification, equations (18a) and (18b) can be written as follows: which is a system of second-order differential equations, where u n+1 (x, y) � u(x, y, t n+1 ) and equations (21a) and (21b) are a system of second-order differential equations.

Fully Discretization in Space.
In this section, spatial derivatives that occur in equations (21a) and (21b) are discretized by modified CTBS DQM over the given domain. After spatial discretization, equations (21a) and (21b) convert into a system of linear equations for each n in the following form: where u n ij � u n (x i , y i ) and α (2) ik and β (2) ik are WCs of 2 nd order partial derivatives w.r.t. x and y.

Implementation of Dirichlet Boundary Conditions.
e Dirichlet BCs given in equation (3) as (α, y, t) � h 1 (y, t), u(β, y, t) � h 2 (y, t), u(x, c, t) � h 3 (x, t), and u(x, δ, t) � h 4 (x, t) can be implemented directly as follows: As a result of applying the BCs on systems (23a) and (23b), the system can be written as follows: Journal of Mathematics e system of equations (18a) and (18b) is a Lyapunov system of equations of the form 6 Journal of Mathematics Also, , ,

Numerical Experiments and Discussion
Under this heading, to check the correctness and competence of the algorithm modified CTBS DQM, two test problems have been considered, which are available in the literature. All the computation work is conducted by using MATLAB 7.0. e following formulas are used for computing maximum absolute error L ∞ , root mean square (RMS) error, and L 2 error, respectively: v(x, y, t) � 3 4 + 1 ICs and BCs are taken from exact solutions (29) and (30). e numerical results are shown with the help of Tables 2  and 3 and Figures 1-4 in form of errors, three-dimension, and contour plots. Convection prevails the flow which causes the errors become larger and larger as we increase the value of Re. L ∞ is smaller than [15] for T � 2.0, Re � 100 with less grid points N � M � 40. e figures show that exact solutions and numerical solutions are well consistent in three-dimensional and contour form. Table 4 shows that, as we increase the values of M and N, the absolute errors decrease which shows the convergence of the method. , (x, y) ∈ D, and BCs,        Table 4 shows that, as we increase the values of M and N, the absolute errors decrease which shows the convergence of the method.

Conclusion
In this study, a modified CTBS DQM and a new algorithm to reveal the computational modeling of 2D coupled Burgers' equations are developed. e proposed algorithm is tested on two benchmark problems appearing in the literature. e main results of this study are summarized as follows: (i) A different technique using modified CTBS functions is presented to determine the WCs of 2D DQM than Lagrange interpolation traditional technique [22]. (ii) CTBS DQ algorithm proposed in [33] has extended for 2D problems in different forms, and it has concluded the algorithm worked nicely for the same problems. (iii) e developed algorithm is better than the DQ algorithms proposed in [31,32,34] due to more smoothness of CTBS functions.
(iv) e presented method leads to quite similar results to those treated in [12,15,17,18] and good accuracy in the case of a small number of grid points. (v) After some modifications, the presented method can be extended to solve 2D or higher-dimensional equations. In this way, it can be used to analyze many other biological, mechanical or physical events, such as reaction, linear diffusion, dispersion, and nonlinear convection.
Comparison of numerical solutions (NSs) and exact solutions (ESs) are given on left and right sides, respectively, in Figures [1-8] for R � 100 and Δt � 0.001.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.