Abstract
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.
1. Introduction
Burgers’ equation involves the convection term, diffusion term, and kinetic viscosity coefficient 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 traffic simulation. Burgers’ equation is as follows [1–6]:with boundary conditions asand initial condition aswhere be the kinetic viscosity. Boundary conditions sometimes are presented as periodic boundary conditions .
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 finite difference method, finite element method, mixed finite element method, characteristics mixed finite element, spectral method, and meshless method; see [1–6] and the references therein.
With the help of Lagrange interpolation, the barycentric rational interpolation method is obtained [7–9]. 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-differential equation. With the further expansion of the application fields, the barycentric rational collocation method has been successfully applied to solve some initial value problems and boundary value problems by Wang et al. [13–15]. The relevant calculation results show the stability advantages and high accuracy of the barycentric rational collocation method. The research of the barycentric rational collocation method for the heat-conduction equation, biharmonic problem, second-order Volterra integro-differential 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. [16–22]. In these papers, error estimation and numerical simulation are given.
The main goal of the present paper is to solve the nonlinear Burgers’ equation with the barycentric rational collocation method. 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.
2. Notations and Barycentric Rational Interpolation
Define the partition of space interval as
Polynomialdenotes the -order Lagrange interpolation with
The barycentric interpolation function is presented aswhere denotes the blending function as follows:
According to the definition of , it can be deduced thatwhere denotes the interpolation weight function as follows:
Through simple derivation, we know
Combining (5)–(11), the barycentric rational interpolation function is presented aswhere weight function is given in (10) and interpolation basis function is defined by
The -order differential function at the mesh-point is obtained as
Its -order differential matrices formulation can be written intowhere
According to definition of in (13), we get the first-order derivative of interpolation basis function as
Combining equations (17)–(19) together, the -order differential recurrence formula of is
For the nonlinear Burgers’ equation with , we partition the space region intoand time interval into
Function is approximated by its barycentric rational interpolation as follows:where
Taking (23) into equation (1), we seewhere is the first-order derivative of the function .
Taking in equation (25), we get
Note that ; after further simplification of equation (26), we knowwhere
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
Through similarly derivation, the discrete scheme of time variable is obtained as
According to equations (27)–(33), we have
Equation (34) can be written into vector form as follows:which can be restated as a simple form:with
Here, operation symbol represents the Kronecker product.
Then, we get the -order differential at the mesh-point as
Its matrices formulation iswhere
The element of the second-order differentiation matrices is defined by
Then, we get the 1-order time differentiation matrix as follows:
Similarly, the 1-order and 2-order space differentiation matrices are obtained:
The -order differential matrix recurrence formula is presented as follows:
3. Convergence Analysis and Error Estimates
Define the error between and 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 , combining (46) with (45), we havewhere
Define
The following lemma has been proved by Berrut et al. in [7].
Lemma 1. For the error defined in (45), if function satisfies certain smoothness conditions on interval , we have
Now, we research the rational interpolation to approximate the function as follows:
Note that the weight function is defined by
Here, parameters represent the space interpolation parameter and time interpolation parameter, respectively.
The error function between and is defined by
Based on Lemma 1, we get the following theorem.
Theorem 1. For the error functional , if , we have
Proof. By equation (53), we haveNote thatCombining equations (55)–(57) together, the proof of Theorem 1 is completed.
Theorem 2. For the error functional defined as (53), if function satisfies certain smoothness conditions on , we have
Proof. By equation (53), we knowCombining equations (56), (57), and (61) together, the error estimate (58) is obtained. The proof of (59) and (60) is similar.
Let be the numerical solution of function as follows:
Theorem 3. Let and , we have
Proof. Following equations (1) and (62), we havewhereAs for the first term in equation (65), we knowThen, we getConsidering the second term of equation (65), we haveThen, we seeTo the third term of equation (65), we getThen, we haveCombining results (68), (70), and (72), the proof is finished.
Remark 1. In the programming of numerical simulation, to deal with the nonlinear characteristic of Burgers’ equation, we adopt the following iteration algorithm:or the Newton–Rapson iteration algorithm.
4. Numerical Experiments
In this section, some numerical experiments with the barycentric rational collocation method are carried out for Burgers’ equation.
Example 1. Consider the following Burgers’ equation :The analysis solution is chosen to beIn Table 1, the errors of the barycentric rational collocation method with Chebyshev nodes with are presented. The absolute errorand relative errorare listed.
We can see from Table 1 that the minimum absolute error can reach and . The minimum relative error can reach and , respectively. The calculation results show that the proposed method has high accuracy feature.
In Table 2, in order to test the convergence rates of space variable in the case of equidistant subdivision, we take the time interpolation parameter with . In Table 3, adopting equidistant subdivision, we take the space interpolation parameter with to test the convergence rates of time variable. In Table 4, in order to test the convergence rate of space variable in the case of Chebyshev nodes, we take the time interpolation parameter with . In Table 5, adopting Chebyshev nodes, we take the space interpolation parameter with to test the convergence rates of time variable. Taking , Figure 1 shows the exact solution, numerical solution, and error with equidistant nodes for the barycentric rational collocation method.
Example 2. Consider the following Burgers’ equation :The analysis solution is set to beIn Table 6, in order to test the convergence rates of space variable in the case of equidistant nodes, we take the time interpolation parameter . In Table 7, adopting Chebyshev nodes, we take the time interpolation parameter with to test the convergence rates of space variable. In Table 8, in order to test the convergence rates of time variable in the case of equidistant nodes, we take the space interpolation parameter with . In Table 9, adopting Chebyshev nodes, we take the space interpolation parameter to test the convergence rates of time variable. Figure 2 shows the exact solution, numerical solution, and error in the case of equidistant nodes for the barycentric rational collocation method with .
Example 3. Consider the following Burgers’ equation :The analysis solution is chosen to beIn Table 10, in order to test the convergence rates of space variable in the case of equidistant subdivision, we take the time interpolation parameter with . In Table 11, adopting equidistant subdivision, we take the space interpolation parameter with to test the convergence rates of time variable. In Table 12, in order to test the convergence rate of space variable in the case of Chebyshev nodes, we take the time interpolation parameter with . In Table 13, adopting Chebyshev nodes, we take the space interpolation parameter with to test the convergence rates of time variable. Taking , Figure 3 shows the exact solution, numerical solution, and error with equidistant nodes for the barycentric rational collocation method.
Remark 2. Numerical experiments using the barycentric rational collocation method for Burgers’ equations show the consistency of the convergence rates with the theoretical analysis. Note that if we further increase the space interpolation parameter and time interpolation parameter , we can get more accurate results. Some numerical results show that the convergence rate is higher than our theorem which is out of our goal of this paper and will be presented in other papers. In the future, we will research the (1 + 2) dimensional and (1 + 3) dimensional Burgers’ equations.
Data Availability
No other data were used in this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of the first author was supported by the Foundation of Shandong Jianzhu University (No. H21010Z), the Development Plan of Youth Innovation Team of University in Shandong Province (No. 2021KJ067), the Shandong Province Soft Science Research Project (No. 2020RKB01671), and the Natural Science Foundation of Shandong Province (Nos. ZR2020ZD25 and ZR2021MF009). The work of the second author was supported by the Special Project for Numerical Forecast Development of China Meteorological Administration (No. GRAPES-FZZX-2021).