In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional 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 time-dependent 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, easy-to-program 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 root-mean-square error are used.
The convection–diffusion, advection–diffusion, or drift-diffusion 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 finite-difference 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 high-order time-stepping methods combined with mesh-free methods is studied for the transient convection–diffusion equation . A Petrov–Galerkin method and Green’s functions are used to solve convection–diffusion problems . A new approach to construct a stable RKPM method for convection-dominated problems is presented in ref. . The space–time least-squares finite element methods are constructed for the advection–diffusion equation by using both linear shape functions and quadratic B-spline shape functions . The particle transport method is developed for solving linear convection problems . Several literatures focus on the investigation of different splines as interpolating function for solving one-dimensional advection–diffusion equations [34,37]. These literatures focus on the one-dimensional convection–diffusion–reaction equations with constant diffusion coefficient. Recently, a high-accuracy adaptive difference strategy is investigated by Zhu and Rui  on 1D convection–diffusion–reaction equation with convection item. It can explain the quenching phenomena of nonlinear singular degenerate problems. Pourgholi et al.  proposed a meshless method using radial basis functions method based on the finite-difference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. These numerical techniques are based on two-level finite difference approximations.
It is well-known 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 one-level approximation, based on the radial basis functions, for the one-dimensional 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 time-dependent 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 one-dimensional linear and nonlinear convection–diffusion problems. Some conclusions are given in Section 5 with some additional remarks.
2 Formulation of the direct radial basis function
To describe the interaction between convection effects and diffusion transports, we can get the general mathematical formulation of convection–diffusion–reaction problem
where is the unknown temperature or scalar quantity, is the convection velocity, is the diffusion, and is the prescribed source term. For linear convection–diffusion equations, and are positive numbers. For nonlinear convection–diffusion equations, one can change into a nonlinear convection term with a nonlinear source term . To seek for the solutions of convection–diffusion–reaction problem equation (1), one should consider the additional initial condition
and boundary conditions
For traditional numerical techniques, equation (1) should be discretized using the finite difference method or integral transform method, which leads to a steady-state equation. Then, the other numerical techniques can be used to get the numerical solutions. This provides a two-level procedure. To obtain a one-level procedure, we propose a direct collocation scheme by using the Gaussian radial basis function (GRBF).
For direct RBF-based collocation methods, the approximate solution can be written as a linear combination of RBFs for the approximation space under consideration for 2D or more higher-dimensional problems. We take the following GRBF for 2D problems as an example ref. 
where is the Euclidean distance between two points and , and is the RBF shape parameter.
However, there is only one space variable for the one-dimensional convection–diffusion problems, the traditional RBF can not be used directly. Here, we construct a simple space–time radial basis function by combining the space variable and time variable as a point for physical domains . More specifically, the interval is uniformly divided into segments firstly with the corresponding finess . For non-uniform cases, the corresponding finess is . The time variable is evenly chosen from the initial time to a final time as with time-step . The corresponding configuration of the space–time coordinate is shown in Figure 1, where stands for the value of space variable , stands for the value of time-variable , and stands for the point . Then, we have the simple anisotropic space–time radial basis function
This can be easily extended to two-dimensional or high-dimensional cases.
3 Methodology for DMM
Based on the definition of space–time radial basis functions, the above-mentioned equations (1)–(3) can be solved directly in a one-level approximation. Thus, the approximate solution of the function has the form
To seek for the unknown coefficients , we can collocate the convection–diffusion equation (1) at internal points, while the initial condition equation (2) and boundary condition equation (3) are collocated at initial collocation points and boundary points, respectively. This procedure yields the following equations:
is known matrix with submatrices
with elements , ,
with elements , , ,
with elements , , ,
with elements , ,
with elements , ,
with elements , ,
with elements ,
with elements ,
with elements , , and
is vectors. This can be solved by the backslash computation in MATLAB codes.
4 Numerical experiments
where is the analytical solution at test points , , and is the numerical solutions at the test points , . is the number of test points on the physical domain. Since the DMM method is stable, the choice of RBF parameter is fixed for all the following cases. The optimal choice of RBF parameter is beyond the scope of our current research. For more details about this topic, we refer readers to ref. [3,26] and references therein.
In the first two examples, we consider two linear cases with governing equation
4.1 Linear example 1
This example considers the following initial condition
and boundary conditions
The corresponding analytical solution is given by
For fair comparison, the dimensionless Péclet number is defined as [1,2]. The coefficients in the governing equation are , , , , , and the corresponding low Péclet number is . For , we compare the DMM results with compactly supported radial basis functions, thin-plate spline radial basis function, and the finite difference method [1,2], detailed results are listed in Table 1. For parameters and with the other parameters fixed as earlier, the corresponding high Péclet number is . Numerical results of the DMM are listed in Table 2. It should be pointed that our time step , which leads to less computations, is larger than the one in ref. . However, the DMM results are more accurate than all results (TPS, FD, CS) presented in ref. .
As is known to all, when the Péclet number is low, the diffusion term dominates. In order to compare the DMM with the cubic -spline collocation method (CFD) in ref. , the corresponding dimensionless Péclet number is defined as . Numerical results of the DMM in terms of average errors are listed in Table 3. From which we can find that the CFD method performs not well for relatively large Péclet numbers even for dense discretization . However, the DMM results are stable and accurate than the CFD . This reveals that the DMM is stable for high Péclet numbers.
4.2 Linear example 2
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 finite-difference approximation of fourth-order and the cubic -spline collocation method .
We consider the convection–diffusion equation (1) with the following initial condition:
and boundary conditions
The analytical solution is given by
For a fair comparison, we use the definition of dimensionless Péclet number in ref. . We choose , , . Numerical results of the DMM are listed in Table 4 with various Péclet numbers for at final time and . We note that our time step , which leads to less computations, is larger than the one 0.001 in ref. . For different values of , we can find that the DMM results are more accurate than the FD results presented in ref.  for Péclet number . For , the results are similar. For Péclet number and , the DMM results are more accurate than the FD results presented in ref. . This reveals that the DMM is more stable than the FD method for various Péclet numbers.
4.3 Nonlinear example 3
In this example, we consider a typical nonlinear convection–diffusion equation
with the exact solution
to measure the performance of the DMM by comparing with the method in ref. . The source term can be derived from the exact solution. The following initial condition and boundary conditions are considered:
We note that for smaller values of , the exact solution varies greatly at . For fair comparison with the method in ref. , we consider and , for our DMM method, respectively. Detailed results are listed in Table 5.
For , the non-uniform scheme in ref.  performances better than the uniform scheme with dense mesh or more. Obviously, the DMM method performs better than these two schemes with sparse mesh or less. It can be seen that the computation accuracy of our DMM is superior to those of the others from the above-mentioned results.
In this study, a new direct meshless scheme is proposed for the one-dimensional 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 time-dependent 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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