Spectral collocation method for convection-di ﬀ usion equation

: Spectral collocation method, named linear barycentric rational interpolation collocation method (LBRICM), for convection-di ﬀ usion (C-D) equation with constant coe ﬃ cient is considered. We change the discrete linear equations into the matrix equation. Di ﬀ erent from the classical methods to solve the C-D equation, we solve the C-D equation with the time variable and space variable obtained at the same time. Furthermore, the convergence rate of the C-D equation by LBRICM is proved. Numerical examples are presented to test our analysis


Introduction
In this study, we consider the convection-diffusion (C-D) equation with constant coefficient: where ( ) [ ] ( ] ∈ × x t a b T , , 0 , and q and s are the constants related to fluid diffusion coefficient and concen- tration, respectively.
There are lots of methods to invest the C-D equation, such as finite difference method [1].Spectral methods [2] have been widely used in lots of scientific engineering area.
In [3], nonlinear C-D equation is studied by the finite-difference lattice Boltzmann model.In reference [4], error estimate of a nonlinear C-D equation by finite element method is presented.In [5], the nonlinear C-D equation was studied and posteriori error estimates were also given.In [6], some spectral and pseudospectral approximations of Jacobi and Legendre type are considered for the C-D equation.In [7], a high-order accurate method for solving the one-dimensional heat and advection-diffusion equations is presented.In reference [8], C-D equation was solved by new formulas of the high-order derivatives of the fifth-kind Chebyshev polynomials.In [9], a transient C-D equation is considered.Time-fractional diffusion equation was solved by Petrov-Galerkin Lucas polynomials in [10].In reference [11], the time-fractional diffusion equation was dealt by the explicit Chebyshev-Galerkin scheme.In reference [12], fractional diffusion-wave equation was dealt by the shifted fifth-kind Chebyshev polynomials.In reference [13], two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional was solved by the Legendre-Chebyshev spectral collocation method.In reference [14], fractional calculus approach was used for oxygen diffusion from capillary to tissues.
In this study, we use the linear barycentric rational collocation method to solve the C-D equation.With the help of convergence rate of interpolation barycentric rational function, both the convergence rate of space and time of linear barycentric rational collocation method for the CD equation can be obtained at the same time.
This article is organized as follows: in Section 2, the differentiation matrices and collocation scheme for the C-D equation are presented.In Section 3, the convergence rate of space and time is proved.At last, some numerical examples are listed to illustrate our theorem.

Collocation scheme for C-D equation
The area m and = τ T n being the uniform partition.However, for the quasi-uniform partition, the second kind of Chebyshev point where and where is the basis function [36].
Then, we change equation (1) into the following: where ( ) ( ) . Then, we have where In order to obtain the time discrete, we take the barycentric interpolation function to the t similarly as x: …, , ij i j i j (7) and then we have where = j n 1, 2,…, .We transform the linear equations into the matrix form as: and where and ⊗ denotes the Kronecker product [32,38].

Convergence and error analysis
We define the error of ( ) u x with ( ) r x as: where Spectral collocation method for convection-diffusion equation  3 where (16) and We first define barycentric rational interpolants as: where and The error of ( ) u x t , with ( ) r x t , mn is defined as: mn i id i i id j jd j j jd Now, we present the following theorem.
Proof.For ( ) ∈ x t , Ωand ( ) e x t , , we have and We reach that By the similarly analysis in Floater and Kai [29], we have ( ) and ( ) Combining ( 23), (24), and (25) together, the proof of Theorem 1 is completed.

□
Corollary 1.For the ( ) e x t , defined in (19), we have Let ( ) u x t , be the analysis solution of (1), ( ) u x t , m n is its approximate value, there holds and m n m n , (29) Based on the aforementioned lemma, we obtain the following theorem.
Theorem 2. Let operator D be defined as (28) and ( ) ∈ f x t , Ω, and we have , . Proof.As where As for the R 1 , we have By the corollary, we obtain Similarly, for ( ) , we have and Spectral collocation method for convection-diffusion equation  5 Combining the identity ( 30), ( 32), (34), and ( 36), the proof of theorem is complete.

□ 4 Numerical examples
Three examples are presented to illustrate our theorem.All examples were performed on personal computer by Matlab 2013a with a (configuration: Intel(R) Core(TM) i5-8265U CPU @ 1.60GHz 1.80 GHz).
Example 1.Consider the C-D equation: with the condition x δ b t t 2 0 In this example, we test the linear barycentric rational with the equidistant nodes., and = β 1 to test our algorithm.Figure 1 shows the error estimate of equidistant nodes, and Figure 2 shows the error estimate of Chebyshev nodes.From Figures 1 and 2, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes, and the barycentric rational interpolation collocation method also has higher accuracy under the condition of equidistant nodes.
In Table 3, we test the linear barycentric rational with the Chebyshev nodes, which that shows the convergence rate is ( ) first given for the space area.In Table 4, for = d 9 2 first given, the convergence rate of times is ( ) , which agrees with our theorem analysis., and = β 1, and the error estimate is given as blow for the equidistant nodes and Chebyshev nodes.
Example 2. Consider the C-D equation:  Spectral collocation method for convection-diffusion equation  7 Its analysis solutions is  , and = β 1 (   , and = β 1 to test our theorem.Figure 3 shows the error estimate of equidistant nodes, and Figure 4 shows the error figure of Chebyshev nodes.From Figures 3 and 4, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes, and the barycentric rational interpolation collocation method also has higher accuracy under the condition of equidistant nodes.
In this example, in order to test our theorem analysis, we have first given = d 9 1 and = t 1 to be the exact solution.Table 6 shows that the convergence rate of equidistant nodes is ( ) O τ d 2 , which agrees with our theorem analysis.In Table 7, in order to test the convergence rate of space, = d 9 2 is first given to be the exact solution, and the convergence rate of times is ( ) O h d 1 , which is one order higher than our theorem analysis.In Table 8, we test the linear barycentric rational with the Chebyshev nodes, and shows that the convergence rate is ( ) and = t 10 first given for the space area.In Table 9, for the space area partition = d 9 1 first given, the convergence rate of times is ( ) , which is one order higher than our theorem analysis.is first given to be the exact solution and the convergence rate of times is ( ) , which is one order higher than our theorem analysis.From this example, the same convergence rate is the same as the linear C-D equation, while the theorem analysis is beyond our goal and will be given in another study.
Figure 5 shows the error estimate of equidistant nodes.From Figure 5, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of equidistant nodes.Spectral collocation method for convection-diffusion equation  11

Conclusion
In this study, (1+1)-dimensional C-D equation has been solved by LBRICM, and the discrete C-D equation can be written to matrix equation using the Kronecker product.For the linear C-D equation, the convergence rate of space and time at the same time is proved with the constant coefficient.However, for the nonlinear C-D equation, the convergence rate is the same.As for (2+1)-dimensional linear and nonlinear C-D equation, magneto-micropolar equations, and fractional C-D equation, we will study in further article to overcome the difficulty of the full coefficient matrix.

Table 1 :
Convergence rate of equidistant nodes with = d 9

Table 2 :
Convergence rate of equidistant nodes with =

Table 3 :
Convergence rate of Chebyshev nodes with = d 9

Table 4 :
Convergence rate of Chebyshev nodes with = d 9

Table 5 :
Convergence rate of long time with =

Table 6 :
Convergence rate of equidistant nodes with = d 9 1 and = t 1

Table 7 :
Convergence rate of equidistant nodes with =

Table 8 :
Convergence rate of Chebyshev nodes with =

Table 9 :
Convergence rate of Chebyshev nodes with = d 9

Table 10 :
Convergence rate of long time with =

Table 12 :
Convergence rate of equidistant nodes with =

Table 11 :
Convergence rate of equidistant nodes with = and we test our algorithm to solve the nonlinear C-D equation.In order to test our theorem analysis, we have first given = d 91 and = t 1 to be the exact solution.Table11shows that the convergence rate of equidistant nodes is ( ) O τ d 2 , which agrees with our theorem analysis.In Table12, in order to test the convergence rate of space, = d 9