1 Introduction

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 [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 convection-dominated problems is presented in ref. [30]. 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 [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 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 [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 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

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. [27]

However, there is only one space variable x 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 x and time variable t as a point ( x , t ) for physical domains [ 0 , L ] × T . More specifically, the interval [ 0 , L ] is uniformly divided into segments firstly 0 = x 0 < x 1 < ⋯ < x n = L with the corresponding finess h = L / n . For non-uniform cases, the corresponding finess is h = max { Δ x j } = max { x j − x j − 1 } ∣ j = 1 n . The time variable is evenly chosen from the initial time t 0 = 0 to a final time t n = T as 0 = t 0 < t 1 < ⋯ < t n = T with time-step Δ t = T / n . The corresponding configuration of the space–time coordinate is shown in Figure 1, where ∘ stands for the value of space variable x , ∗ stands for the value of time-variable t , and ⊛ stands for the point ( x , t ) . Then, we have the simple anisotropic space–time radial basis function

Figure 1

Configuration of the space–time coordinate.

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].

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 C ( x , t ) has the form

A 11 with elements L φ j ( P i , P j ) , i , j = 1 , 2 , … , N I ,

A 12 with elements L φ j ( P i , P j ) , i = 1 , 2 , … , N I , j = N I + 1 , … , N I + N 1 ,

A 13 with elements L φ j ( P i , P j ) , i = 1 , 2 , … , N I , j = N I + N 1 + 1 , … , N ,

A 21 with elements φ j ( P i , P j ) , i = N I + 1 , … , N I + N 1 , j = 1 , 2 , … , N I ,

A 22 with elements φ j ( P i , P j ) , i = N I + 1 , … , N I + N 1 , j = N I + 1 , … , N I + N 1 ,

A 23 with elements φ j ( P i , P j ) , i = N I + 1 , … , N I + N 1 , j = N I + N 1 + 1 , … , N ,

A 31 with elements ∂ φ j ( P i , P j ) ∂ t , i = N I + N 1 + 1 , … , N , j = 1 , 2 , … , N I ,

A 32 with elements ∂ φ j ( P i , P j ) ∂ t , i = N I + N 1 + 1 , … N , j = N I + 1 , … , N ,

A 33 with elements ∂ φ j ( P i , P j ) ∂ t , i = N I + N 1 + 1 , … , N , j = N I + N 1 + 1 , … , N , and

4 Numerical experiments

To compare with the previous literatures, we consider using the absolute error, root-mean-square 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

5 Conclusions

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.