In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the onedimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove timedependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easytoprogram and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and rootmeansquare error are used.
The convection–diffusion, advection–diffusion, or driftdiffusion equations have been playing a significant role in many engineering applications. The energy can be transformed inside a physical system due to the convection and diffusion processes to describe physical phenomena.
For a large variety of problems in every subjects, it is almost impossible to get the analytical solutions for changing in time and transport processes [23,24]. Numerical approximations are alternative to the analytical solutions for convection–diffusion equations. In literatures, there are several numerical techniques for solving the convection–diffusion equations [33,41]. These numerical techniques are based on the finitedifference approximations [25,32], integral transform methods [28,29], Monte Carlo simulation [4,5], variational iteration methods [6,7, 8,9,10], and meshless method using radial basis functions [11, 12,13,14, 15,16,17, 18,19,20].
The behavior of highorder timestepping methods combined with meshfree methods is studied for the transient convection–diffusion equation [31]. A Petrov–Galerkin method and Green’s functions are used to solve convection–diffusion problems [21]. A new approach to construct a stable RKPM method for convectiondominated problems is presented in ref. [30]. The space–time leastsquares finite element methods are constructed for the advection–diffusion equation by using both linear shape functions and quadratic Bspline shape functions [22]. The particle transport method is developed for solving linear convection problems [44]. Several literatures focus on the investigation of different splines as interpolating function for solving onedimensional advection–diffusion equations [34,37]. These literatures focus on the onedimensional convection–diffusion–reaction equations with constant diffusion coefficient. Recently, a highaccuracy adaptive difference strategy is investigated by Zhu and Rui [47] on 1D convection–diffusion–reaction equation with convection item. It can explain the quenching phenomena of nonlinear singular degenerate problems. Pourgholi et al. [43] proposed a meshless method using radial basis functions method based on the finitedifference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. These numerical techniques are based on twolevel finite difference approximations.
It is wellknown that the radial basis function methods are attractive in numerical simulation due to their simple, flexible, and truly meshfree features. In this study, we propose a direct meshless method with onelevel approximation, based on the radial basis functions, for the onedimensional linear and nonlinear convection–diffusion problems. This is fulfilled by considering the time variable as a normal space variable. There is no need to remove the timedependent variable during the whole solution process. Under this scheme, we can solve the convection–diffusion problems in a direct way.
The structure of this paper is organized as follows. Followed by Section 2, we introduce the formulation of the direct radial basis function (DMM) with space–time distance. Section 3 presents the methodology for convection–diffusion problems under initial conditions and boundary conditions. Section 4 examines several linear with different Péclet numbers and nonlinear problems. Several numerical examples are presented to validate the accuracy and stability of the proposed algorithm for onedimensional linear and nonlinear convection–diffusion problems. Some conclusions are given in Section 5 with some additional remarks.
To describe the interaction between convection effects and diffusion transports, we can get the general mathematical formulation of convection–diffusion–reaction problem
For traditional numerical techniques, equation (1) should be discretized using the finite difference method or integral transform method, which leads to a steadystate equation. Then, the other numerical techniques can be used to get the numerical solutions. This provides a twolevel procedure. To obtain a onelevel procedure, we propose a direct collocation scheme by using the Gaussian radial basis function (GRBF).
For direct RBFbased collocation methods, the approximate solution can be written as a linear combination of RBFs for the approximation space under consideration for 2D or more higherdimensional problems. We take the following GRBF for 2D problems as an example ref. [27]
However, there is only one space variable
In the literature, there is a product model of a space–time radial basis function [42], which was introduced by Myers et al. [39,40],
The other types of definitions of radial or non radial space–time radial basis functions can be found in ref. [35,36].
Based on the definition of space–time radial basis functions, the abovementioned equations (1)–(3) can be solved directly in a onelevel approximation. Thus, the approximate solution of the function
To compare with the previous literatures, we consider using the absolute error, rootmeansquare error (RMSE) [45,46], and average absolute error (AAE) defined as follows:
In the first two examples, we consider two linear cases with governing equation
This example considers the following initial condition
The corresponding analytical solution is given by
For fair comparison, the dimensionless Péclet number is defined as
t 





AAE  

0.1  TPS  0.0834  0.0451  0.0244  0.0132  0.0069  0.0000 
FD  0.0836  0.0454  0.0244  0.0133  0.0073  0.0001  
CS  0.0848  0.0452  0.0244  0.0132  0.0077  0.0003  
DMM  0.0833  0.0451  0.0244  0.0132  0.0072 


0.5  TPS  0.1376  0.0744  0.0402  0.0216  0.0113  0.0001 
FD  0.1384  0.0757  0.0413  0.0225  0.0123  0.0008  
CS  0.1407  0.0757  0.0407  0.0223  0.0131  0.0010  
DMM  0.1374  0.0744  0.0402  0.0218  0.0118 


1  TPS  0.2570  0.1390  0.0751  0.0404  0.0211  0.0002 
FD  0.2587  0.1420  0.0778  0.0425  0.0232  0.0018  
CS  0.2631  0.1420  0.0768  0.0421  0.0246  0.0023  
DMM  0.2568  0.1389  0.0752  0.0407  0.0220 

t 





AAE  

0.1  TPS  0.1001  0.0782  0.0611  0.0478  0.0373  0.0000 
FD  0.1006  0.0788  0.0616  0.0481  0.0469  0.0013  
CS  0.1005  0.0782  0.0611  0.0477  0.0372  0.0001  
DMM  0.1000  0.0779  0.0607  0.0473  0.0368 


0.5  TPS  0.1650  0.1289  0.1007  0.0787  0.0615  0.0001 
FD  0.1663  0.1316  0.1038  0.0817  0.0858  0.0042  
CS  0.1661  0.1297  0.1011  0.0787  0.0613  0.0004  
DMM  0.1649  0.1285  0.1001  0.0780  0.0607 


1  TPS  0.3083  0.2409  0.1882  0.1470  0.1149  0.0001 
FD  0.3106  0.2460  0.1948  0.1545  0.1626  0.0087  
CS  0.3102  0.2424  0.1893  0.1478  0.1152  0.0009  
DMM  0.3081  0.2100  0.1870  0.1457  0.1135 

As is known to all, when the Péclet number is low, the diffusion term dominates. In order to compare the DMM with the cubic
Pe 






FD  1/16 




1/32 





1/64 





1/128 





DMM  1/16 




Once the Péclet number increases, i.e., the convection term completely dominates over the diffusion term. In order to investigate problems with the far higher Péclet numbers, we compare the DMM results with the compact finitedifference approximation of fourthorder and the cubic
We consider the convection–diffusion equation (1) with the following initial condition:
The analytical solution is given by
Pe  DMM  FD  DMM  FD  DMM  FD 

x  1,000  1,000  10,000  10,000  20,000  20,000 
0.25 






0.50 






0.75 






1.00 


0 



1.25 






1.50 






1.75 






In this example, we consider a typical nonlinear convection–diffusion equation
We note that for smaller values of

DMM  Uniform  Nonuniform 

8 

—  — 
16 

—  — 
32 

—  — 
64 



128  — 


256  — 


For
In this study, a new direct meshless scheme is proposed for the onedimensional linear and nonlinear convection–diffusion problems. The present numerical procedure, in which the time variable is considered as normal space variable, is based on the Gaussian radial basis function. There is no need to remove timedependent variable during the whole solution process. Numerical results for several typical examples show that the proposed method is better than some other numerical methods given in the recent literature in terms of solution accuracy, stability and efficiency for the linear convection–diffusion equation with different values of Péclet number. These results lead us that the proposed method can successfully be used to nonlinear problems with accurate numerical results.
This work was supported by the Natural Science Foundation of Anhui Province (Project No. 1908085QA09) and the University Natural Science Research Project of Anhui Province (Project No. KJ2019A0591 & KJ2020B06).
Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.
Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request.
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