Barycentric Rational Collocation Method for Burgers’ Equation

In this article, barycentric rational collocation method is introduced to solve Burgers’ equation. The algebraic equations of the barycentric rational collocation method are presented. Numerical analysis and error estimates are established. With the help of the barycentric rational interpolation theory, the convergence rates of the barycentric rational collocation method for Burgers’ equation are proved. Numerical experiments are carried out to validate the convergence rates and show the efficiency.


Introduction
Burgers' equation involves the convection term, di usion term, and kinetic viscosity coe cient whose characteristic is same as the structure of the Navier-Stokes equation without the stress term. It describes the phenomena such as dispersion in porous media, weak shock propagation, heat conduction, acoustic attenuation in fog, compressible turbulence, gas-dynamics, continuous stochastic processes, and even continuum tra c simulation. Burgers' equation is as follows [1][2][3][4][5][6]: with boundary conditions as u(α, t) ϕ 1 (t), u(β, t) ϕ 2 (t), t> 0, and initial condition as where ] > 0 be the kinetic viscosity. Boundary conditions sometimes are presented as periodic boundary conditions u(α, t) u(β, t) 0.
In view of the universality of Burgers' equation in describing lots of important physical phenomena, many numerical methods were introduced to solve it such as the nite di erence method, nite element method, mixed nite element method, characteristics mixed nite element, spectral method, and meshless method; see [1][2][3][4][5][6] and the references therein.
With the help of Lagrange interpolation, the barycentric rational interpolation method is obtained [7][8][9]. A rational interpolation scheme with equidistant and special distributed nodes has been proposed by Floater and Hormann [10]. Compared with Lagrange interpolation, the barycentric rational interpolation has the advantages of stability. Abdi et al. [11,12] have used the barycentric rational collocation method to solve Volterra and Volterra integro-di erential equation. With the further expansion of the application elds, the barycentric rational collocation method has been successfully applied to solve some initial value problems and boundary value problems by Wang et al. [13][14][15]. e relevant calculation results show the stability advantages and high accuracy of the barycentric rational collocation method. e research of the barycentric rational collocation method for the heat-conduction equation, biharmonic problem, second-order Volterra integro-di erential equation, third-order two-point boundary value problem, beam force vibration equation, telegraph equation, and incompressible Forchheimer flow in porous media has been presented in recent papers by Li et al. [16][17][18][19][20][21][22]. In these papers, error estimation and numerical simulation are given. e main goal of the present paper is to solve the nonlinear Burgers' equation with the barycentric rational collocation method. O(h d 1 − 1 + τ d 2 ) error estimates are proved. Numerical experiments are carried out to show the convergence rates. Remaining part of the paper is structured as follows. In Section 2, the barycentric rational interpolation formula is given. In Section 3, convergence analysis of the barycentric rational collocation method for the nonlinear Burgers' equation is presented. Section 4 reports some test examples to show the accuracy, effectiveness, and efficiency.

Notations and Barycentric
Rational Interpolation Polynomial denotes the d-order Lagrange interpolation with y k � y(x k ) e barycentric interpolation function R(x) (d � 0, 1, . . . , m) is presented as where μ i (x) denotes the blending function as follows: According to the definition of μ i (x), it can be deduced that where ϖ k denotes the interpolation weight function as follows: rough simple derivation, we know i+d k�i i+d j�i,j≠k Combining (5)- (11), the barycentric rational interpolation function R(x) is presented as where weight function ϖ j is given in (10) and interpolation basis function R j (x) is defined by e s-order differential function at the mesh-point x i is obtained as Its s-order differential matrices formulation can be written into where y (s) � y (s) 0 , y (s) 1 , . . . , y (s) n , s � 1, 2, . . . , According to definition of R j (x) in (13), we get the firstorder derivative of interpolation basis function R j (x) as 2 Journal of Mathematics Combining equations (17)- (19) together, the s-order differential recurrence formula of D (s) ij (s � 1, 2, . . .) is For the nonlinear Burgers' equation with Function u(x, t) is approximated by its barycentric rational interpolation as follows: where u j (t) � u x j , t , j � 0, 1, . . . , m.
Taking (23) into equation (1), we see where _ u j (t) is the first-order derivative of the function u j (t).
Note that R j (x i ) � δ ij ; after further simplification of equation (26) where Combining equations (26)-(28), the matrix form is presented as Further, matrix equation (29) can be rewritten into a simple vector form as follows: where rough similarly derivation, the discrete scheme of time variable t is obtained as According to equations (27)-(33), we have (j � 0, 1, . . . , n) Equation (34) can be written into vector form as follows: which can be restated as a simple form: with (37) Here, operation symbol ⊗ represents the Kronecker product.
en, we get the s-order differential at the mesh-point x i as Its matrices formulation is where en, we get the 1-order time differentiation matrix as follows: Similarly, the 1-order and 2-order space differentiation matrices are obtained: e s-order differential matrix recurrence formula is presented as follows: Journal of Mathematics

Convergence Analysis and Error Estimates
Define the error between u(x) and R(x) as follows: According to the error theory of interpolation, it is well known that In the light of the definition of barycentric rational interpolation function R(x), combining (46) with (45), we have where Define e following lemma has been proved by Berrut et al. in [7].
Now, we research the rational interpolation R m,n (x, t) to approximate the function u(x, t) as follows: (51) Note that the weight function ω ij is defined by Here, parameters d 1 and d 2 represent the space interpolation parameter and time interpolation parameter, respectively.

e error function E(x, t) between u(x, t) and R m,n (x, t) is defined by
Based on Lemma 1, we get the following theorem.

Theorem 1. For the error functional
Proof. By equation (53), we have

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.
Let u(x m , t n ) be the numerical solution of function u(x, t) as follows: Proof. Following equations (1) and (62), we have      Table 4: Absolute errors and convergence rates in the case of Chebyshev nodes with time interpolation parameter d 2 � 9 for Example 1.  Journal of Mathematics 7 As for the first term A 1 in equation (65), we know (67)         Journal of Mathematics

en, we get
(68) Considering the second term A 2 of equation (65), we have A 2 � u(x, t)u t (x, t) − u x m , t n u t x m , t n � u(x, t)u t (x, t) − u x m , t u t (x, t) + u x m , t u t (x, t) − u x m , t u t x m , t + u x m , t u t x m , t − u x m , t u t x m , t n + u x m , t u t x m , t n − u x m , t n u t x m , t n � A 21 + A 22 + A 23 + A 24 (69) en, we see To the third term A 3 of equation (65)