Abstract
In this work, a CMFS method based on the analogy equation method, the radial basis function and the method of fundamental solutions for linear and nonlinear convection-diffusion equations in anisotropic materials is presented. The analog equation method is utilized to transform the linear and nonlinear convection-diffusion equation into an equivalent one. The expressions of the homogeneous solution and particular solution are derived by utilizing the radial basis function approximation and the method of fundamental solutions, respectively. By enforcing the desired solution to satisfy the original convection-diffusion equation with boundary conditions at boundary and internal collocation points yield a nonlinear system of equations, which can be solved by using the Newton-Raphson iteration or the Picard method of iteration. The error convergence curves of the proposed meshless method have been investigated by using different globally supported radial basis functions. Numerical experiments show that the proposed CMFS method is promising for anisotropic convection-diffusion problems with accurate and stable results.
1 Introduction
Partial differential equations (PDEs) are generally utilized in understanding and modeling of a considerable lot of realism matters show up in applied science and material science. The PDE models are utilized in numerous fields, for example, plasma physics, hydrodynamics, finance, biology and nonlinear optic [1–3]. It is pointed that it is hard to tackle nonlinear problems most of the cases, especially analytically. Numerous strategies have been developed by the researchers for the numerical solution of complex problems (see [4–6] and the references therein). Among these techniques, one of the most attractive group of techniques is radial basis function (RBF) based techniques.
The RBF techniques are attractive in numerical simulation thanks to their simple, flexible, and truly meshfree features. These techniques have been successfully applied to diverse problems in a simple-to-implement fashion. The popular RBF-based numerical methods include the Kansa’s method [7, 8], the method of fundamental solution (MFS) [9–12], the boundary knot method [13–15], the modified method of fundamental solutions [16–18] which have been well-developed and applied to a variety of boundary value problems.
In previous literatures, the polynomial RBFs, thin plate spline, Gaussians, and Multiquadrics are often used [19–21]. It is noted that the traditional RBF-based schemes are indirect and global in the sense that the expansion coefficients are used as the basic variables in the numerical solution procedure, while the global RBF interpolation leads to the full matrix. Roughly speaking, the RBF-based approaches can be classified as two types. The first category uses RBFs to approximate the particular solution of a partial differential equation (PDE) of interest, and the homogeneous part is obtained by means of numerical methods, like the boundary element method, the MFS, or the boundary knot method. The second category is the domain-type collocation methods. According to this approach, the RBF expansion is utilized directly for the unknown solution, and the collocation satisfies the governing equation and boundary conditions. The Kansa’s method is a typical domain-type RBF approach. In this paper, we will develop a RBF-based technique of the first type to investigate anisotropic materials.
Anisotropic materials, characterized by varied material properties along different directions, are ubiquitous in nature and difficult to be analyzed. If non-linear property is included, the problems become even more complicated. Previously, Shin and Elman [22] studied the effect of various element discretization strategies and iteration algorithms for nonlinear convection-diffusion problems with variable velocity in isotropic materials. Torsten [23] used the anisotropic streamline-diffusion finite element method to analyze homogeneous convection-diffusion problems with dominant convection. Onyejekwe [24] applied Green element method to 2D transient convection-diffusion problems with linear reaction and variable velocity. The lattice Boltzmann method is proposed for general nonlinear anisotropic convection-diffusion equations [25,26]. The anisotropic nonlinear convection-diffusion equations are also investigated by the finite volume method [27], the finite element method [28], the finite difference method [29], the virtual element method [30]. Shang et al. [31] proposed a discrete unified gas kinetic scheme for a general nonlinear convection-diffusion equation. Cao and Zhang studied a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields [32].
Based on the above-mentioned investigations, we aim to apply the MFS, in combination with the RBF and the analog equation method (AEM) [33], to analyze anisotropic nonlinear convection-diffusion problems. First, the AEM is utilized to transform the PDE into an equivalent one. Then, the expressions of the homogeneous and particular solutions are derived by utilizing RBF approximation and the MFS, respectively. Finally, enforcing the desired solution to satisfy the original PDE with boundary conditions at boundary and internal collocation points yield a nonlinear system of equations, which can be solved by using the Newton-Raphson iteration or the Picard method of iteration. Here, we will focus on the construction of Picard method of iteration, which should be carefully constructed to reach convergence.
The structure of this paper is arranged as follows. In Section 2, we give the depiction of nonlinear convection-diffusion problems in anisotropic media. Followed in Section 3, the detailed processes are derived to construct the proposed CMFS. Numerical investigation and results analysis are carried out in Section 4. Section 5 concludes this paper and provides some comments on the CMFS.
2 Nonlinear Convection-Diffusion Problems in Anisotropic Materials
Consider an open-bounded domain where represent problem’s dimension. Assuming that is bounded by a piecewise smooth boundary which is consist of numerous sufficiently smooth segments in the Liapunov sense. The space coordinate is utilized to denoted position of an arbitrary point. The description of convection-diffusion problems in anisotropic media can be expressed as partial differential equationsubjected to Dirichlet boundary conditionand Neumann boundary condition related boundary fluxwhere denotes a velocity vector and is the desired field. is the constant material tensor. Particularly, the material constant tensor for 2D cases is given by
The boundary flux is defined as , where represent components of the unit outward normal vector to the boundary . stands for the internal forcing function, and are specified values on the Dirichlet and Neumann boundary, respectively. For a well-posed problem, .
It is observed that the smaller the determinant of the more asymmetric and anisotropic are the field and flux vectors, the more difficult it is to get an accurate numerical solution. Specially, if is a diagonal matrix with completely equivalent diagonal elements, Eq. 1 reduces to an isotropic convection-diffusion equation. In general, when material tensor or velocity vector is dependent on the unknown filed Eq. 1 shows nonlinearity. Here, we restrict our attention to research such nonlinear cases, velocity is considered as a function related to the desired field u, that is, .
3 The CMFS Meshless Method
For the previous nonlinear problems, the proposed CMFS method is based on the combination of the AEM, globally supported RBF approximation and the MFS. Detailed processes are given below.
3.1 The Analog Equation Method
The AEM, which was improved by Burlon et al [34], was first proposed by Katsikadelis for the solution of nonlinear problems. Using the analog equation method, Eq. 1 can be converted into a Poisson type equation. Suppose is the sought solution of Eq. 1, which is a continuously differentiable function with up to two orders in . Apply the Laplacian operator to , we can derive
If the source distribution is known, then the solution of Eq. 1 can be produced by solving the linear Eq. 3 under the same boundary conditions (2a)-(2b), namely, Eq. 3 is equivalent to Eq. 1.
Using to the principle of superposition, the solution of linear PDE Eq. 3 can be written in the form of the sum of the particular solution and the homogeneous solution , which is given as
Accordingly, and should satisfyandrespectively. Furthermore, we must note that the particular solution satisfying Eq. 5 does not need to satisfy boundary conditions, so it is not unique and must be coupled with the homogeneous solution and related boundary conditions. However, our aim is just to obtain the approximating expression rather than detailed numerical methods.
3.2 Radial Basis Function Approximation to the Particular Solution
This step is to derive the particular solution by RBF approximation. There are two schemes to fulfill this procedure. The standard approach to obtain the particular solutions is to integrate an operator by means of selected RBF. This scheme is just suitable to some simple operators and RBFs which is mathematically reliable. Another approach is a reverse differential process, which has no restriction on certain operators and RBFs, but the selected RBFs must be continuously differentiable with high orders. Here, the introduction of AEM replaces the original complicated operator with a simpler Laplacian operator, so it is feasible to evaluate the particular solution using the integrating process. To this end, the fictitious term introduced in Eq. 3 can be approximated bywhere is the number of interpolation points inside the domain . denote coefficients to be determined, and represents the radial basis function.
Similarly, it is reasonable to express the particular solution asif the following relationshipholds, where is a corresponding particular basis function depending on the radial basis function and can be obtained by repeated integration.
The accuracy and efficacy of the interpolation depend on the choice of the radial basis function , which should provide an accurate approximation to b so that Eq. 9 can be derived analytically. During the past several decades, polynomial RBFs, MQs and TPS interpolation have got hot attention from the science and engineering communities. They have been investigated for Laplacian operator in and . It is noted that the MQs converge exponentially, TPS in , and traditional , where is minimum separation distance [35]. As is known to all, the better the approximation properties of RBFs corresponds with the worse conditioning. For example, MQs are spectrally convergent whereas their condition number increases exponentially when the data density increases. Additionally, the selection of shape parameter in MQs is also an important factor to approximation and its small variation may cause severe differences to solutions. Thus, we consider the implementation of polynomial RBFs and MQs approximation in this paper.
3.3 The Method of Fundamental Solutions for the Homogeneous Solution
Before introducing the MFS [36, 37], it is necessary to give the definition of the fundamental solutions. The fundamental solution for the Laplacian operator is defined to satisfyin an infinite domain, where denotes the Dirac delta function which goes to infinity for the case of and equals to zero elsewhere. The detailed expression of for Laplace equation is
The fundamental idea of the MFS is to place a virtual boundary outside the domain interested. Here, to obtain a weak solution of Laplace Eq. 6, collocation points and source points are distributed on the physical boundary and the virtual boundary, respectively. Furthermore, suppose that there is a virtual source load at each fictitious source point. The difference between the physical and virtual boundaries means that the homogeneous part at arbitrary field points in the domain or on the physical boundary can be constructed by a linear combination of fundamental solutions in terms of fictitious sources , that is,which exactly satisfies Eq. 6.
The proper usage of the MFS must concern three problems. The first case is the number of collocation points distributed on the physical boundary. However, too many collocation points may aggravate the ill-conditioned matrix. The virtual boundary shape is another important aspect. Theoretically, the virtual boundary shape can be arbitrarily chosen in the calculation. In practical computation, the virtual boundary shape is usually selected as a circle or similar shape to the actual boundary to keep algorithm versatile [38]. For example, for the rectangular domain, the rectangular or circular virtual boundaries can be used (see Figure 1).
FIGURE 1
The location of the fictitious source points is also an interesting issue. It has been investigated in several literatures [39–41]. In this paper, we consider a new way to find out the proper location of the virtual boundary, a ratio parameter is introduced and can be defined as follows
For example, in Figure 1, if the length and height of the rectangle domain are and , respectively, the location of circular and similar rectangular virtual boundary can be measured by diameter , and correspondingly. From the computation perspective, mathematical accuracy will decrease if the distance between the virtual and physical boundary turns out to be close, because of the singular disturbance of the fundamental solutions. Then again, in case when the source points are far away from the physical boundary then round-off error in floating point arithmetic may be a major issue in such case, the coefficient matrix of the system of equations is closely to zero [38]. Therefore, parameter is commonly chosen to be in the range of 1.8–4.0 and 0.6–0.8 for internal and external problems respectively. Unless otherwise specified, a circular virtual boundary will be employed and parameter is used to determine its location in this paper.
3.4 The Construction of Solving Equations
According to the above process, the solution to Eq. 3 can be expressed aswhich is also the solution of Eq. 1. Differentiating Eq. 14 yieldswhere and , .
There are two different approaches to determine the unknowns
and
for nonlinear cases. If
Eqs 14–
16directly satisfy the governing
Eq. 1at
interpolation points in the domain
and corresponding boundary conditions (2a)-(2b) at
boundary collocation points, a linear or nonlinear system of
equations can be obtained. The Newton-Raphson iteration [
42] is a good choice to solve the nonlinear system. However, if we first linearize the nonlinear term, and then make
Eqs. (14)-
(16)satisfy
Eq. (1)and boundary conditions (2a)-(2b) at related collocation points, the Picard method of iteration can be carried out. Compared to the Newton-Raphson iteration, the Picard method of iteration doesn’t require computation of Jacobian matrix. However, the linearization of the nonlinear term in the Picard method of iteration is a key step. This process should be carefully considered to reach convergence. In the paper, the Picard method of iteration is constructed as follows.
(1) Assume an initial guess at interpolating points
(2) During an iteration , given ,
(a) Linearization of Eq. 1:
(b) Using the AEM-RBF-MFS to produce the following linear solving equations
(c) The unknown coefficients and can be obtained by solving the above system of linear equations.
(d) Evaluating field values at interpolating points by means of
(e) Convergence criteria: if , exit loop; else, let for next iteration.
(3) Once the iteration converges, Eq. 14 is used to evaluate quantities at arbitrary point in the domain and on the physical boundary.
4 Numerical Experiments
In this section, the convergence and accuracy of the CMFS are numerically examined by solving anisotropic nonlinear convection-diffusion problems. Since the Kansa’s method is a traditional RBF-based approach, comparisons are made between the CMFS and the Kansa’s method. To measure the accuracy of the approximation, the relative error , average relative error are defined as belowwhere and denote the analytical and numerical results, respectively. is the total number of tested points. In our computation, tested points with number are uniformly distributed in the square domain.
In order to investigate the condition number and convergence of the proposed CMFS, the linear anisotropic convection problem is tested in the first case, and then, nonlinear Burger’s equation and nonlinear anisotropic convection-diffusion equation are subsequently examined.
4.1 2D Linear Anisotropic Convection-Diffusion Problems
We first consider the linear anisotropic convection-diffusion equation in a square domain where and in the constant tensor and velocity vector is given by .
The analytical solution shown is given asfor this case, which is also used to derive the Dirichlet boundary condition.
For the selection of interpolating points, different opinions always exist. Figure 2 shows the average relative error of polynomial and MQs whether interpolating points M include boundary collocation points or not. Form Figure 2, we can see the MQs is better than .
FIGURE 2
For definite internal points number , the solution accuracy of the CMFS with different RBFs and its condition numbers with increasing boundary points are shown in Figure 3. It is observed from Figure 3 that accuracy does not show significant improvement when the number of interpolating points maintains. Besides, Figure 4 displays the solution accuracy of the CMFS with different RBFs and its condition numbers with increasing boundary points in solving anisotropic convection-diffusion problems when the number of boundary collocation points is set. We can scrutinize higher convergence rate of the solution accuracy with increasing internal interpolating points than convergence rate with increasing boundary points. Compared to conventional polynomial RBFs, the MQs generally leads to huge condition number of with corresponding high solution accuracy . As mentioned in pervious literatures, the better the approximation properties of the RBFs, the worse the conditioning number [43]. In addition, we note that the solution accuracy by using MQs is better than using polynomial RBFs with two or three orders of magnitude. The choice of the RBFs shape parameter is investigated in some literatures [44].
FIGURE 3
FIGURE 4
4.2 Nonlinear Inviscid Burger’s Equation
In this case, the nonlinear form for steady-state situation is consideredwhere the variable represents a velocity term. A particular solution for this problem is , which is also the exact solution when imposed as a boundary condition. An peanut irregular domain is considered for this case (Figure 5). The initial guess of is selected as one.
FIGURE 5
Figure 6 shows the convergence curves of the CMFS using the Picard method of iteration with different RBFs. All convergence curves can be found with the increase of internal collocation points . Meanwhile, we also observe that MQ can reach more accuracy when a certain boundary points are employed to implement RBF approximation, instead of pure interior points. However, this phenomenon seems to disappear for and . Furthermore, the usage of pure internal interpolating points improves accuracy in the case of small M. The similar fact that MQ have better accuracy than polynomials is observed, this is eliminated in this case.
FIGURE 6
4.3 Nonlinear Anisotropic Convection-Diffusion Problem
Consider a 2D anisotropic convection-diffusion problem depicted bywith and in the same square domain as Example 4.2. The analytical expression is used with Dirichlet boundary only, the right-handed source function is chosen to be
Here, the similar Picard iteration scheme is employed and the initial guess of also is set to one. Figure 7 displays the average relative error curve at 1,000 test points against the different interpolating collocation schemes. It is found that whether the boundary collocation is included or not, the CMFS convergence curve converges stably and quickly in solving the nonlinear anisotropic convection-diffusion problem than the polynomial RBFs and . And the solution accuracy of the CMFS is averagely three orders of magnitude larger than the polynomial RBFS and . However, the instability for RBF should be observed when the RBF approximation involves the usage of the boundary collocation points.
FIGURE 7
5 Conclusion
In this paper, the CMFS, which is composed by the analog equation method, radial basis function approximation, and the method of fundamental solutions, is applied to the nonlinear anisotropic convection-diffusion problems. Numerical results reveal the efficiency and stability of the CMFS for the tested three cases. In a word, the proposed CMFS has the following features:
1) For linear cases, the approach is just one-step simple scheme. Meanwhile, no inverse of a matrix is involved.
2) The simple fundamental solution of Laplacian operator is employed, rather than one of the original PDE.
3) There are simple process and theoretical basis, so it is easy to program.
4) The related Picard iteration process is developed for nonlinear cases in isotropic and anisotropic media, respectively.
5) The proposed method is a truly meshfree method and no integrals are needed.
6) It can be easily extended to solve anisotropic un-homogeneous problems with variable parameter or the other problems [45].
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
LZ supported found of the manuscript; FW developed the conceptualization, and wrote the manuscript; JZ worked on the mathematical development, performed the numerical analysis; SN and TN analyzed the data, and wrote the manuscript. All authors have read and agree to the published version of the manuscript.
Funding
This work is partially supported by the National Natural Science Foundation of China (Project No.U1965110) and the Natural Science Foundation of Anhui Provincial (Project No. 2008085MA11).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
radial basis function, meshless method, convection-diffusion problems, nonlinear partial difference equation, boundary value problem
Citation
Zhang L, Wang FZ, Zhang J, Wang YY, Nadeem S and Nofal TA (2022) Novel Numerical Method Based on the Analog Equation Method for a Class of Anisotropic Convection-Diffusion Problems. Front. Phys. 10:807445. doi: 10.3389/fphy.2022.807445
Received
02 November 2021
Accepted
03 March 2022
Published
04 April 2022
Volume
10 - 2022
Edited by
Horacio Sergio Wio, Institute of Interdisciplinary Physics and Complex Systems (CSIC), Spain
Reviewed by
Yang Liu, Inner Mongolia University, China
Choonkil Park, Hanyang University, South Korea
Hussein A. Z. AL-bonsrulah, Iran University of Science and Technology, Iran
Updates
Copyright
© 2022 Zhang, Wang, Zhang, Wang, Nadeem and Nofal.
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*Correspondence: J Zhang, zj801106@163.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.