Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

: This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are ﬁrst approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefﬁcients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.


Introduction
Many phenomena in physics, biology and engineering can be modelled mathematically by partial differential equations (PDEs). The Burgers equation is one of the most important PDEs to be surveyed in the recent years by many researchers [1,2]. This equation describes various kinds of phenomena in plasma physics, solid state physics, optical fibers, fluid dynamics, chemical kinetics, non-linear acoustics, gas dynamics, traffic flow, etc.
Also, the generalized Burgers-Fisher equation is one of the most important classes of non-linear PDEs which has appeared in several categories of applications, such as shockwave formation, turbulence, heat conduction, sound waves in viscous medium, and some other fields of applied branches of science and engineering [3]. Moreover, The Burgers-Huxley equation has been considered to be an evolution equation that describes nerve pulse propagation in biology from which molecular CB properties shall be computed. The generalized Burgers-Huxley equation was investigated to describe the interaction between reaction mechanisms, convection effects, and diffusion transport [4].
Since an analytical in a closed-form solution is generally unavailable for non-linear PDEs, numerical methods are widely used for solving them. There are some effective numerical methods to solve PDEs, especially for the Burgers equation. In [5], a comprehensive review of some techniques is presented. Berger and Kohn in [6] used the med refinement method. Budd et al. in [7] applied mesh movement. Soheili et al. used a moving-mesh PDE (MMPDE) approach [8]. In [9], Ramadan et al. suggested a method based on collocation of septic B-splines over finite elements for numerical the convergence of the CSC method is analyzed. Next, the two-dimensional Burgers equation is introduced in Section 5 and the CSC method is applied to solve the equation. Then, Section 6 contains numerical examples to solve the Burgers equations in both cases of one and two-dimensional. Also, the figures of errors (associated with the proposed method) are depicted in some cases which confirm the efficiency of our suggested numerical scheme. Finally, the paper is concluded with a reasonable conclusion.

Burgers Equation
Three important types of Burgers equations are as follows: 1. For a field V(., .) and diffusion coefficient (or viscosity, as in the original fluid mechanical context) ε, the general form of viscous Burgers equation is as follows (1)

The Burgers-Fisher equation is a non-linear PDE of second order of the form
where α(.) and β(.) are given functions. It plays an important role in various fields of gas dynamics, traffic flow, physics applications, financial and applied mathematics [23].

The Burgers-Huxley equation is as follows
where α, β, δ, γ and η are given constants. The Burgers-Huxley as a non-linear PDE describes the interaction between reaction mechanisms, convection effects, and diffusion transports [23].
In this paper, we represent the Burgers Equations (1)-(3) as the following general form The time initial and space boundary conditions (in Dirichlet form) for Burgers Equation (4) are usually given as follows

Implementing the CSC Method
Here, the CSC method for Burgers Equation (4) with conditions (5)-(7) is presented. The CSC method [32][33][34] is one of the most efficient numerical methods to solve continuous-time problems. Recently, some researchers have applied it to solve special problems (see [35][36][37]). One of the most important advantages of CSC method in comparison with other approximate methods is the high degree of accuracy that CSC approximations offer. Also, the CSC underlying polynomial space spanned by Chebyshev orthogonal polynomials on the interval [−1, 1] with respect to a weight In the CSC method, we use some points on the interval [−1, 1] to discretize the problem. Hence, we first transform the variables of Equation (4) to this interval by the following Therefore, Burgers Equation (4)-(7) converted into the following form where ).
To discretize system (9), we use the CGL points on [−1, 1] which are defined by the following relations where they are the roots of (1 −t 2 ) dT N dt and T N (.) is the Chebyshev polynomial of order N. It should be noted that the Chebyshev polynomials are expressed by and it is easy to show that For interpolating in the CSC method, the following Lagrange polynomials are used and we have In the CSC method, to approximate the solution of Burgers Equation (9), we use the following polynomial interpolation To express the derivative U N t (·, ·), U N x (·, ·) and U N xx (·, ·) in terms of U N (·, ·) at the node pointst k , we can use the matrix multiplication D = (D kj ) and get where D = D · D = (D kj ),D kj = ∑ N l=0 D kl D lj , k, j = 0, 1, . . . , N. In fact, multiplication by matrix D transforms a vector of the state variables at the CGL points to the vector of approximate derivatives at these points. Now, by relations (14) and (15), the system (9) can be converted into the following system of algebraic equations Here, by solving above system with respect to (ā N pk ; p, k = 0, 1, . . . , N), we can obtain continuous and pointwise approximate solutions (13) and (14), respectively.

The Convergence of the Method
In this section, first we give definition of the modulus of continuity and then analyze the convergence of the presented method.
Some important modulus of continuity can be defined as Now, assume that B 2 is a unit circle in R 2 . We say that a continuous function f (·, ·) onΩ admits W(·) as a modulus of continuity, if the following value is finite With C 1 W (B 2 ) we denote the space of all functions f (·, ·) on B 2 with continuous first-order partial derivatives, and let it is endowed with the following norm Next, define It can be proved that is a Banach space (for more details see [38]). At follows, we show the space of all polynomials of total degree at most 2N onΩ by Pol(N, N,Ω), i.e., where c 1 = f (·, ·) 1,W and c 0 is a constant that depends on W(·), but independent of N.
Proof. The proof is a result of Theorem 2.1 in [38].
To prove the existence of solutions of the system (17), we convert it in the following form where N is sufficiently big and W(·) is a given modulus of continuity. Since lim N→∞ . . , N) for system of algebraic inequalities (25) is a solution for system of algebraic Equation (17) when N tends to infinity.
We assume that ψ has bounded and continuous derivatives with respect to its arguments. Hence, there exists a constant M such that Now we will show that the system (25) has at least one solutionā N .

Theorem 2.
Let U(·, ·) be a solution for system (9) where U(·, ·) is in C 1 W (Ω). Then there is a positive integer K such that for any N ≥ K, system (25) has a solution as which satisfies where L is a positive constant independent of N.

Numerical Examples
In the following examples, we use the Levenberg-Marquardt method (a quasi-Newton method) for FSOLVE command in MATLAB software to solve the algebraic system (17). We calculate L 2 and L ∞ errors as follows where x j , j = 0, 1, . . . , N, are the collocation points,t ∈ [−1, 1] is a given point and U(·, ·) and U N (·, ·) are the analytical and approximate solutions, respectively. Also, the absolute error can be obtained by (2), where T 0 = −0.2, T 1 = 0, a = −1, b = 0, α(t) = 24 and β(t) = −48 for t ∈ [T 0 , T 1 ]. Also, we assume that the boundary conditions are given by

Example 1. Consider the Burgers-Fisher Equation
and the initial condition is as follows Then the exact solution is We gain the numerical results by the CSC method at t = −0.1, −0.05, −0.04, −0.035, −0.03 for N = 30 × 30 which are shown in Table 1. We observe that our numerical results are better than the results of MMPDE methods [8], which they are obtained for ∆t = 10 −6 and ∆x = 1 60 (or N = 200, 000 × 60). In Table 1, L 2 errors are presented. In Figures 1 and 2, we show the approximate solution and absolute error, respectively. In Figure 3, we represent the exact and approximate solutions for t = −0.1, −0.05 and −0.03. We also illustrate L 2 errors in Figure 4. , , and the initial condition .
By these, the exact solution is as follows We solve the system (17) according to this example. Table 2 shows the L 2 errors at t = −0.05, −0.025, 0, 0.025 and 0.05 for CSC method and MMPDE methods [8]. Our numerical results are satisfied N = 10 × 10 (or equivalently ∆x = 0.1 and ∆t = 0.01) and results of the MMPDE methods are with ∆x = 1 40 and ∆t = 10 −4 (or equivalently N = 1000 × 40). In Figures 5, 6 and 7, we illustrate the approximate solution, absolute error and the L 2 error, respectively.    (3) with α = 1, δ = 2 and γ = 0. Therefore, it can be written as follows

Example 3. Consider the Burgers-Huxley Equation
where T 0 = 1, T 1 = 10, a = 0 and b = 1. Also, consider the boundary conditions and the initial condition where 0 < c 0 < 1. Hence, the exact solution is given by .
We take c 0 = 0.5 and β = 0.01. Table 3 shows the L ∞ errors by using CSC method and given methods in [9,12,15,16] for t = 2, 4 and 6 and N = 9 × 9. In Figures 8 and 9, we show the approximate solution and absolute error, respectively. Also, in Figure 10, we represent the exact and approximate solutions for t = 2, 6 and 10. Moreover, Figure 11 shows the L ∞ error. Table 3. Comparison of the L ∞ error for Example 3.
The numerical results of our method and other methods [12,16] are displayed in Table 4 for different values of t. The L ∞ error results for this example is depicted in Table 4 along with the comparison of the error computed by the present method and other methods. In Figures 12 and 13, the approximate solution and absolute error are shown, respectively. Also, in Figure 14, we represent the exact and approximate solutions for t = 2, 6 and 10.     (1), where T 0 = 1, T 1 = 5, a = 0, b = 8 and ε = 0.5. The initial condition for the current problem is

Example 5. Consider the Burgers Equation
and the boundary conditions are U(t, 0) = 0 , Here, we have the following analytical solution .
In Figures 15 and 16, we show the approximate solution and absolute error, respectively. Also, Figure 17 illustrates the comparison between the exact solution and numerical solution given by the proposed method. Also, we compare the L ∞ and L 2 errors which are computed by the present method and other methods [14] in Table 5. We can observe that the results of CSC for N = 30 × 30 are better than the results of method of Inan and Bahadir [14] for N = 320 × 40, 000.    Example 6. By considering the two-dimensional Burgers Equation (41) with T = 1 and ε = 0.1, 0.2, 0.5 and 1, the initial condition is as follow [30,31] U(x, y, 0) = 1 and the boundary conditions are , U(1, y, t) = 1 U(x, 0, t) = 1 U(x, 1, t) = 1 ) . (60) By these considerations, the exact solution is U(x, y, t) = 1 1 + exp( (x+y−t) 2ε ) .
We calculate approximate solutions and absolute errors in different ε = 0.1, 0.2, 0.5 and 1 with N = 10. In Figures 18 and 19, we observe the approximate solutions and absolute errors with ε = 0.1, respectively. Also, Figures 20 and 21 illustrate the numerical solution and absolute error given by the presented method for ε = 0.1. The approximate solutions and absolute errors with ε = 0.5 and 1 are expressed in Figures 22-25, respectively. Moreover, we computed the L ∞ errors for various ε at t = t 0 which are shown in Table 6.

Conclusions
In this article, we used the Chebyshev spectral collocation method to get numerical solutions for different types of one and two-dimensional Burgers equation. We analyzed the convergence of the CSC method by using the concept of module of continuity and compared the obtained approximate solutions with those of other methods. We showed that the CSC method has very high accuracy and it is more precise with respect to the other numerical methods. Our investigations can be used in the three-dimensional case and we prepare these investigations as a future article.
Author Contributions: The authors contributed equally to this work.
Funding: This research was funded by the National Natural Science Foundation of China Projects grant number [11601240,11771214].

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: CSC Chebyshev Spectral Collocation CGL Chebyshev-Gauss-Lobatto