Abstract
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. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The 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. The technique can be extended for various multidimensional Burgers’ equations after some modifications.
1. Introduction
In this paper, the authors considered the following dimensionless form of two-dimensional (2D) coupled Burgers’ equation:with initial conditions (ICs),and Dirichlet boundary conditions (BCs),where is Laplace operator, and are velocity components to be determined. Also, are known smooth functions and is the Reynolds number with density , viscosity , characteristic length L, and .
The nonlinear convection-diffusion model is simply represented by Burgers’ equation [1]. This famous equation describes the flow theory through a shockwave moving in viscous liquid [2], phenomena of turbulence [3], and various other kinds of phenomena in aerodynamics.
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 [5–8], 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) [18–23] have been developed. In recent years, new meshless methods [24–26] for various types of Burgers’ equations have been developed.
In Lagrange interpolation-based DQMs [18–22], 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. The techniques lead to correct results with insignificant and errors.
2. 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 as follows:where and are unknown coefficients weighting the first and second derivatives, respectively, and , 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, 29–35] and the references therein) to determine these coefficients. In this study, we determine WCs with the use of CTBS functions after some modifications.
2.1. Cubic Trigonometric B-Spline Functions
In this section, we mesh the solution domain into subintervals with the help of knots such that is a uniform partition with step length .
Now, the piecewise CTBS basis functions over the uniform mesh are defined as follows [36, 37]:where
The basis over the region is formed by the set
Every CTBS covers four elements. Now, with the help of Table 1, we have tabulated the values of and its derivatives as follows:
2.2. 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 , are linearly independent. On the solution domain , these functions create a family of basis functions.
2.3. Weighting Coefficients for Modified Cubic Trigonometric B-Spline Differential Quadrature Method
Now, substitute the modified functions , into equation (4). The matrix form of the equation is as follows:where is coefficient matrix:
The matrix , and at , are as follows:
Furthermore, with the help of Thomas algorithm WCs, 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 .
2.4. 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 as by adopting step length and in and direction, respectively. This modified technique helps to estimate the 1st order partial derivatives of at a point as follows:where is WCs for the 1st order derivatives w.r.t. . Similarly, are coefficients w.r.t. .
In order to compute the 2D WCs, we can define the functions , as in equation (10). Furthermore, take the test functions as . Now, with the help of the axioms of vector space and substituting the value of into equations (14) and (15), we have
Furthermore, applying the well-known algorithm “Thomas algorithm” and proceeding with the same methods as in the case of equation (11), the solutions of the systems give the value of and . In 2D case, the WCs in higher-order derivatives can be considered as follows:where and are WCs for order partial derivatives w.r.t. and , respectively.
3. Numerical Algorithm for Two-Dimensional Coupled Burgers’ Equation
In this section, the numerical algorithm is developed in the following sections.
3.1. Semidiscretization in Time
Applying forward difference on time derivatives and weighted average on spatial derivatives, we havewhere step length in time direction, and . The nonlinear term is linearized in the following manner:with ICsand prescribed BCs (3).
After simplification, equations (18a) and (18b) can be written as follows:which is a system of second-order differential equations, where and equations (21a) and (21b) are a system of second-order differential equations.
3.2. 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 in the following form:where and are WCs of 2nd order partial derivatives w.r.t. and .
3.3. Implementation of Dirichlet Boundary Conditions
The Dirichlet BCs given in equation (3) as 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:
The system of equations (18a) and (18b) is a Lyapunov system of equations of the formwhere
Also,
Equations (19a) and (19b) are the coupled Lyapunov system, first solved for and then solved simultaneously for by developing code in MATLAB 7.
4. 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. The following formulas are used for computing maximum absolute error , root mean square (RMS) error, and error, respectively:where are approximate and exact solutions, respectively, and .
Problem 1. As the first problem, consider 2D Burgers’ equations (1a) and (1b). The exact solutions over the domain is generated by the Hopf-Cole transformation [12, 15, 17] and obtained asICs and BCs are taken from exact solutions (29) and (30). The 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. is smaller than [15] for with less grid points . The 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.
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Problem 2. Consider 2D Burgers’ equations (1a) and (1b) over the computational domain with the ICs [12],and BCs,The exact solutions of the problem are given byTables 5 and 6 show , RMS, and errors for different values of Re and time, while Figures 5–8 show a comparison of numerical and exact solution in three-dimensional form. Convection prevails the flow which causes the errors become larger and larger as we increase the value of Re. The figures show that exact 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.
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5. Conclusion
In this study, a modified CTBS DQM and a new algorithm to reveal the computational modeling of 2D coupled Burgers’ equations are developed. The proposed algorithm is tested on two benchmark problems appearing in the literature. The 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)The developed algorithm is better than the DQ algorithms proposed in [31, 32, 34] due to more smoothness of CTBS functions.(iv)The 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 .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.